A MODEL FOR EVALUATING CONTROL
OF UNSTEADY LEADING EDGE
VORTEX.FLOWS
By
STEVEN DWIGHT ROBERTS
Bachelor of Science
Oklahoma State University
Stillwater, Oklahoma
1992
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
MASTER OF SCIENCE
July, 1997
A MODEL FOR EVALUATING CONTROL
OF UNSTEADY LEADING EDGE
VORTEX FLOWS
Thesis Approved:
Dean ofthe Graduate College
PREFACE
This study was conducted to provide new knowledge and tools for investigating
control of unsteady leading edge vortex flows typically found on highly swept aircraft
planforms. Methods for controlling the unsteady strength and spatial characteristics ofthe
vortices in the flowfield using leading edge flaps were modeled and compared to existing
experimental data. The model has also provided a method for investigating new ideas in
control ofunsteady separated vortical flows.
I sincerely thank my committee, Drs. Andrew S. Arena, David G. Lilley, and Frank
W. Chambers for their guidance and support in the completion of this research.
Furthennore, I thank the School of Mechanical Engineering and the Oklahoma Space
Grant Consortium for this research opportunity and their financial support.
I would like to express my thanks to my family for their encouragement and
understanding in difficult times.
III
TABLE OF CONTENTS
CHAPTER 1 1
1. 1 What performance criteria has dictated the use of separated vortical flows? 1
1. 1. 1 Higher velocities 1
1.1.1.1 Examples 1
1.1.1.2 Consequences ofhigher velocity with more efficiency 2
1.1.2 Controllability I Maneuverability 2
1.1.2.1 Landing regime 2
1.1.2.1.1 Consequences 3
1.1.2.1.2 Examples 3
1.1.2.2 Maneuvering 4
1.1.3 Solutionmusing separated vortical flows 4
1.2 How are separated vortical flows generated? 5
1.3 Why is control ofLeading Edge Vortex Flows needed? 5
1.4 Control methodology of Leading Edge Vortex Flaps 6
1.5 Basis for Tool 8
1.5. 1 Experimental Techniques for Modeling Wing Rock 8
1.5.2 Computational Techniques for Modeling Wing Rock 9
1.5.2.1 Navier Stokes and Euler Techni,ques 9
IV
1.5.2.2 InV'lSC'dl, Potentt',al.Techniqu.,es , 10
1.6Rati0 nate tior Develop'mg a.n InV'"ISC"l dMadel 11
CHAPTER 2 , 13
2.1 Leading Edge Vortex Flow Control for Wing Rock 13
2.1.1 Systems Approach , 14
2.1.2 Aerodynamic Control 16
2.1.2.1 Geometric Control through Geometric Modifications 18
2.1.2.2 Pneumatic Control 25
CHAP'fER 3 : , 34
3.1 Survey of Inviscid, Potential Flow Modeling 34
3.2 Solution ofMathematical Model ,. 42
3.2.1 Definition ofGoverning Equations , 42
3.2.1.1 Unsteady Conservation ofMass 42
3.2.1.2 Slender Wing Assumption , 43
3.2.1.3 Conical Assumption , , 43
3.2.1.4 Boundary Conditions 44
3.2.2 Flowfleld Solution 45
3.2.2.1 Singularity Solution Superposition 45
3.2.2.2 Flow Tangency and Influence Coefficients ,.. 49
3.2.2.3 Kutta Condition , 50
3.2.2.4 Kelvin Condition ., Sl
3.2.2.5 Zero Force S3
v
3.2.2.6 Steady Convergence 56
3.2.3 Extension ofModel to Unsteady Motion 57
3.2.3.1 Rigid Body Equation in Roll 58
3.2.3.2 Unsteady Boundary Conditions 58
3.2.3.3 Unsteady Zero Force 59
3.2.4 Unsteady Vortex Flow Control.. 59
CHAPTER 4 61
4. t Static Model Validation 61
4.2 Dynamic Model Validation 66
4.3 Static Vortex Flap Actuation 74
4.4 Dynamic Vortex Flap Actuation and Control 79
4.5 Projected Control Methodologies 85
CHAPTER 5 87
5. 1 Conclusions 87
5.2 Recommendations 90
REFERENCES 92
APPENDIX 96
Vl
LIST OF FIGURES
Figure 3. 1 Delta Wing Cross Sectional Geometry with Vortex Flaps 60
Figure 4.1 Static Lateral Primary Vortex Variation 62
Figure 4.2 Static Normal Vortex Position Variation 63
Figure 4.3 Static Primary Variation of Vortex Strength 64
Figure 4.4 Static Sectional Roll Coefficient Variation 65
Figure 4.5 Conical Pressure Distribution 66
Figure 4.6 Wing Rock Amplitude Envelope 67
Figure 4.7 Experimental Wing Rock Time History 68
Figure 4.8 Computational Wing Rock Time History 69
Figure 4.9 Dynamic Lateral Vortex Position 70
Figure 4.10 Dynamic Variation of Vortex Strength 71
Figure 4.11 Dynamic Normal Vortex Position 71
Figure 4.12 Computational Roil Moment Hysteresis versus Roll Angle for a
Steady State Cycle of Wing Rock 73
Figure 4.13 Experimental Roll Moment Hysteresis versus Roll Angle for a Steady
State Cycle ofWing Rock 73
Figure 4.14 Leading Edge Vort,ex Flap Geometry 74
Figure 4.15 Static Ron Moment Versus Roll Angle for Two Flap Angle
Combinations 75
Figure 4.16 Lateral Vortex Position Variation with Flap deflection 76
VII
I '
Figure 4.17 Normal Vortex Position Variation with flap deflections and roll angle 77
Figure 4.18 Normal Force wilth 6=30.0 degrees 78
Figure 4.19 Normal Force with 8=45.0 degrees 79
Figure 4.20 Wing Rock Suppression 80
Figure 4.21 Flap Deflection History During ControL. 81
Figure 4.22 Roll Moment Time History During Control 81
Figure 4.23 Dynamic Lateral Vortex Position during Control 82
Figure 4.24 Dynamic Normal Vortex Position during Control 83
Figure 4.25 Dynamic Vortex Strength during Control _.. 84
Figure 4.26 Roll Moment versus Roll Angle Stability Curve 85
Vlli
NO:MENCLATURE
CI == roll moment
Cl' == sectional roll moment
CFD == computational fluid dynamics
Cp == coefficient of pressure
cr == root chord length
Ix == inertia in roll of the delta wing
K == feedback gain constant
Laero == total roll moment due to aerodynamic effects
m == total number ofpanels
q == velocity vector
qbody == velocity vector of the body
qoo == freestream velocity vector
t == time
t* == nondimensional time = tVocJcr
up == local velocity on a panel in the x direction
Uv == local velocity on a panel in the x
direction due to a vortex
Vn == velocity of the body nonnal to the panel in
local coordinates
wp == local velocity on a panel in the z direction
Wv == local velocity on a panel in the z direction
due to a vortex
a == angle of attack
8 == flap angle
rl == total vorticity ofthe left primary vortex
r r == total vorticity of the right primary vortex
A == sweep angle
y == vortex strength / panel length
<1> == flowfield potential
¢J == roll angle
¢ == roll rate
~. == roll acceleration
(j == source strength / panel length
IX
CHAPTER 1
INTRODUCTION
1. 1 What performance criteria has dictated the use of separated vortical flows?
In attempting to control unsteady leading edge vortex flows, it is important to
understand which performance criteria have driven the design of the vortex flows. Ever
since the first flight at Kitty Hawk., aircraft designers have been striving to expand the
flight envelope and performance of aircraft. Performance advances such as higher cruise
velocities and greater maneuverability have been gained, but not always without expense.
1.1.1 Higher velocities
Many designs have attained higher cruise velocities by sacrificing aerodynamic
efficiency. The inefficiencies are reflected largely in drag penalties and thus the fuel
economy ofan aircraft.
1. 1.1. 1 Examples
The associated tradeotIs may have been acceptable when the technology was first
demonstrated in aircraft such as the McDonnell FlOl Voodoo, General Dynamic F102
Delta Dagger, or the Convair BS8 Hustler; however, the inefficiency of these
technologies is no longer acceptable. For example, one of the main design criteria of
1
2
Lockheed Martin's F22 aircraft is supercruise, the ability to achieve supersonic flight
without the aid of an afterburner.
1.1.1.2 Consequences ofhigher velocity with more efficiency
Aircraft designers today rely heavily on swept and delta wing geometries to
maintain aerodynamic efficiency for a number of reasons. The benefits of swept wing
geometries discovered in 1935 by Busemann, Betz, and Prandtl are lower wave drag at
supersonic cruise speeds. Building on their efforts, they later show that swept wing and
delta wings designs keep the local chordwise flow subsonic. By maintaining locaUy
subsonic flow, swept wings also experience a smoother increase in drag in the transonic
region. The overall benefit of using sw,ept and delta wings was in alleviating shock wave
inefficiencies in the transonic and supersonic regions.
These designs, while·· solving some of the difficult aspects of high subsonic and
supersonic flight, create other problems. While a design can be optimized for cruise
performance, often the lift to drag performance at lower subsonic speeds is poor due to
swept wings having lower aspect ratios. The effect of a lower aspect ratio is to increase
induced drag. Therefore, lower lift to drag ratios require higher landing velocities to
maintain control of the aircraft.
1.1.2 ControUability / Maneuverability
1.1.2.1 Landing regime
At the lower speeds maneuverability and controllability issues become even more
important, especially when lower speeds are associated often with proximity to the ground
3
as in the taking off and landing phases of flight. These designs create other problems also.
In the case of swept and delta wings, the same design criteria which lead to more efficient
cruise performance often diminishes safety for two reasons. First, swept wing planforms
can generate unrecognized aerodynamic characteristics which result in controllability and
safety problems in "off..design" regions. Secondly, because of having been optimized for
cruise performance, the lift to drag performance at lower subsonic speeds is poor due to
swept wings having lower aspect ratios. For a constant aircraft weight, lower lift to drag
ratios require either higher landing velocities to maintain lift and/or higher angles of attack
to control the aircraft.
1.1.2.1.1 Consequences
So far, designs fulfilling each one of these criteria have disadvantages. Both result
in higher landing speeds that can result in crashes with higher kinetic energy increasing the
chance for loss of life and aircraft. Furthermore, high angles of attack have associated
controllability problems. Aircraft aerodynamics in lower speed, "offdesign", and high
angle of attack regimes can be unsteady and result in unsteady vehicle motion.
1.1.2.1.2 Examples
This fact is demonstrated by an old phenomenon known as wing rock which is an
uncommanded lateral limit cycle oscillation. NASA's High Angle of Attack F18
research Vehicle (HARV) demonstrated how unsteady aerodynamics at high angles of
attack could result in uncommanded roll mode fluctuations. The fluctuations can be limit
cycle oscillations as in the case of the NASA's HARV F18. However, the motion need
not be a limit cyclel
. The unsteady aerodynamics can result in divergent motion as on the
"
4
X31. Even during steady state limit cycle oscillations on the HARV F18, the amplitude
ofthe motion can be slightly fluctuating about a steady state amplitude.
1.1.2.2 Maneuvering
Fighter maneuvers although usually not at lower altitudes do not occur at
supersonic cruise velocities. There£ore, lift performance suffers for the same reason as in
landing phases of flight. A smaller aspect ratio creates less lift for a given angle of attack;
rate of climb diminishes as a result. Maneuvering problems where the ability to track an
adversary is important are more difficult. The maneuvering envelope in many cases is
diminishedto remove the possibility ofuncommanded vehicle motion.
At the very least, controllability problems during landing phases along with
maneuverability or tracking problems during flight increase the work load on the pilot if
the maneuvering envelope is not diminished.
1.1.3 Solution in using separated vortical flows
The essential need for safety during take off and landing phases of flight along with the
desir,e for increased maneuverability has demonstrated the benefits of using separated
vortical flows. Separated vortical flow improves the LID ratio at subsonic speeds as well
as the maneuverability of the delta wing aircraft at moderate angles of attack.
Performance is improved by the vortices over the wing creating additional low pressure
regions. As angle of attack increases, the vortices' strength increases providing a
decreasing pressure over the wing49
. The integrated effect being additional lift.
5
1.2 How are separated vortical flows generated?
Separated flows for the most part are generated using sharp leading edges of wings
and smaller secondary vortex generators. Viscosity causes the freestream flow impinging
on the leading edge of the wing or vortex generator to separate. The shear layer
emanating from each sharp leading edge coalesces into a coiled shear layer. At low to
moderate angles of attack these vortices maintain a fairly stable position and strength. As
the angle of attack increases, the strength of the vortices increases also. However, with
the advent of high angle of attack maneuvering, these same vortices develop a hysteresis
in position and strength.
Vortex flow can also be generated using turbulent separation conditions in the
boundary layer. By designing a pressure gradient in the boundary layer inconsistent with
turbulent separation the flow can remain attached. If desired, the flow can be made to
separate.
1.3 Why is control ofLeading Edge Vortex Flows needed?
The hysteresis in these methods of vorticity generation often pose new problems
because the maneuverability of the aircraft is coupled with the unsteady aerodynamics.
The problem then becomes one of how to control these separated vortical flows so that
the aircraft has desirable handling qualities in all flight modes while increasing the
6
maneuverab~lity of the aircraft itself Control of these separated vortical flows will allow
the exploitation of lift and moment to increase the maneuverability ofthe aircraft.
Under certain conditions the hysteresis in the unsteady aerodynamics can lead to
erratic motion in roll and yaw of the aircraft. However, aircraft which have highly swept
planforms can exhibit a selfinduced oscillatory dynamic roll mode of motion known as
"wing rock,,49. Even with a highly swept planfonn, wing rock is still very dependent upon
specific geometric factors. Wing rock is characterized by a buildup to a limit cycle
oscillation which is independent ofinitial conditions.
Control of the separated vortices is fundamental in problems such as wing rock.
Vortex control is critical to designs of proposed civilian supersonic aircraft that will
operate at high angles of attack during approach and landing to attain a low airspeed.
Tactical aircraft which routinely operate at high angles of attack during combat maneuvers
are also limited in performance by the onset of wing rock oscillations. In order to expand
the operating envelope of these types of aircraft, the problem of dynamically controlling
the vorticity distribution through separated vortices must be understood.
,1.4 Control methodology ofLeading Edge Vortex Flaps
Investigation of all of the many methods that could dynamically control leading
vortex flows is nothing short of daunting. The methods available for investigation of
vortex flows are the traditional experimental methods along with the computational fluid
dynamic (CFD) methods. Both methods yield a great amount of detailed data while
requiring a great amount of effort to glean general trends for a wide range of parameters.
7
Problems arise with studying dynamic problems experimentally with outside
factors becoming important. Factors oft·en considered are whether the motion being
modeled is the being reproduced accurately. Simplifications to the experimental apparatus
may limit investigations to studying one degree of freedom at a time. Then synthesizing
the data back to get a feel for how the multiple degree offreedom of problem is behaving
is difficult. Experimental methods provide a good feel for the actual situation if an
experiment is designed properly.
With the difficulty in obtaining information experimentally, often it is difficult to
design an experiment to investigate only the fundamental parameters. The phenomenon
being studied may be a function of several parameters of which most cannot be directly
measured i.e. the state of the experiment is not completely observable. Add the
complexity of measuring an unsteady flowfield and the design of the experiment becomes
just as complicated as the phenomenon being researched.
Likewise, computational methods such as Navier Stokes solvers can yield vast
amounts of data. Just with even a steady flow solution, questions concerning meshing of
the computational domain can impede a general search to validate new concepts. If the
flow solution is then coupled with the equations of motion for an aircraft, the amount of
data to be understood is overwhelming. Since the boundary conditions are changing in a
dynamic solution of a flow field, the meshed domain also changes. Remeshing a
computational domain at every time step slows a solution procedure immensely.
Predicting any dynamic motion that may be produced by varying the strength and position
of vortices is certainly intractable if a wide range of parameters are being investigated
using CFD methods. However, once the range of parameters has been narrowed, CFD

8
methods provide v,ery detailed itnfonnation applicable to detail design. At the preliminary
design, important larger trends can be obscured by the quantity of data.
1.5 Basis for Tool
During the preliminary stage of studying the feasibility of several or many
promising ideas for controlling leading edge vortical flows, only as much data as needed
to distinguish one idea over another is needed. Once a distinction is made between the
ideas, the more feasible options can be researched in greater detail. But a full study of
each of the preliminary ideas leads to a test matrix size that is intractable a.t best. By
refining the test matrix and studying only the more promising ideas, a solution to a
problem can be found and understood sooner. In order to refine a large test matrix, a tool
is needed that is much faster than either experimental of computational research efforts
are. Distinctions between preliminary ideas are often found in the primary characteristics
of the flow field. Therefore, a tool is needed that is flexible enough that it can be used
with all of the ideas being studied, faster than traditional methods, but does not necessarily
need to be as accurate. These criteria can be accomplished by using simplifications to the
flowfield model that will reveal trends in the implementation ofthe new designs within the
specific limitations ofthe model.
1.5.1 Experimental Techniques for Modeling Wing Rock
A new tool certainly could be developed from a simplification of existing
techniques or a completely new method could be developed that would meet this criteria.
Experimental techniques are difficult to simplify since much of the data acquisition
9
techniques tend to be very tedious. Also, accurate, detailed models must be constructed
for each idea to be studied. Furthermore, measurements of a dynamic flowfield are
difficult to obtain due to constraints of the test apparatus, often times. For example, a
single laser doppler anemometer cannot measure unsteady vorticity since vorticity must be
differentiated from the velocity flow field. Primarily, the desired result.s from the new tool
are a simpler method for reducing the feasibility test matrix.
1.5.2 Computational Techniques for Modeling Wing Rock
Computational methods on the other hand are easier to simplify. Assumptions and
observations can be made which simplify the basis for the model. The implementation of
the model is reduced before a solution is sought. These same assumptions and
observations can refine the search for a solution method. Therefore, by using simplifying
assumptions in computational methods, a more efficient solution can be realized.
1.5.2.1 Navier Stokes and Euler Techniques
Since separated vortical flowfields are being studied, the flowfield is assumed to be
inviscid. This eliminates the stress gradient terms from the Navier Stokes equations
leaving only the convective acceleration terms along with the time derivative terms. From
continuity, the divergence of the velocity field can be rewritten in tenns of a potential
function if the flowfield is assumed be irrotational. The continuity equation then becomes
Laplace's governing equation. Ifa solution of an unsteady form of Laplace's equation is
implemented that is fairly quick, then the requirements for a new tool are met.
Computation techniques in modeling the flow phenomenon ofwing rock have been
10
accomplished by the numerical solution of the Navier Stokes equations, Euler solution
techniques, and potential theory. Numerical simulation of unsteady delta wing flows at
constant roll rates using Navier Stokes equations have been implemented by Gordnier and
Visbal2
• Chaderjian3 has used a NavierStokes simulation code to obtain data for static and
forced dynamic motion cases. These methods are able to match experiment almost
exactly, but even on the most advanced computers, these methods are costly in CPU time.
1.5.2.2 Inviscid, Potential Techniques
Potential flow models for steady delta wing flow fields have been used by Brown
and Michael4
, Mangler and Smiths, and Konstadinopoulos, et al6 More recently, Arena
and Nelson1 present experimental studies along with computational models that show that
the limit cycle oscillation ofthe wing rock phenomenon can be captured by modeling only
the primary physics ofthe aerodynamic characteristics. Arena's model assumes an inviscid
flow field in which all the vorticity is concentrated into two leading edge vortices. The
inviscid assumption for unsteady delta wing flows for angles of attack where vortical
breakdown is not present has been suggested by Arena and others. Their suggestion is
based upon experimental investigation since a large angle of attack region exists where
wing rock is pr,esent but vortex breakdown is not seen. 8 Also, slender wing theory along
with a conical flow field assumption was used to simplify Arena's model to improve
computational time. This assumption is based on and justified by experimental results for
delta wings with no breakdown present. To use the conical flowfield assumption, the
properties of the flowfield must be functions primarily of the lateral and normal
coordinates of the wing. This is true for a steady flow field, but the boundary condition
for the unsteady motion proves the conical assumption mathematically invalid. The
11
boundary condition for a rotating wing or normal velocity of the body is not constant
along conical rays, but of rays parallel to the axis of rotation. Again, experimental
evidence shows that conical flow qualitatively holds for flowfield for the unsteady case;
however, the flow is only locally conical in the unsteady case instead ofglobally conical as
in the steady case. A consequence of locally conical flow is that a solution is found for a
particular chord station and is then linearly scaled for the other chord stations.
These assumptions allow the three dimensional flow field to be approximated using
a two dimensional model. The model qualitatively captured all characteristics observed in
experiment including unsteady behavior of Ct, vortex position, and Cpo Presently, the
physics causing wing rock has been obtained so that methods that suppress or exploit the
wing rock phenomenon can be developed.
1.6 Rationale for Developing an Inviscid Model
The primary motivation for developing an inviscid model is to systematically
investigate a variety of control schemes for controlling unsteady separated leading edge
vortical flows on delta wings. In searching for solutions that will allow dynamic control, a
great number of possible configurations exist that may by successfuL Experimental
investigation of all of the many possibilities that exist even with one configuration makes
the problem practically intractable. Computational investigation requires a significant
inv,estment in computational time. Navier Stokes simulations can take hundreds of hours
to obtain a single cycle of motion. To study wing rock suppression, for example, between
50 and 100 cycles would be needed to capture the transient, steady state, and control
suppression. This translates into an enormous cost in time with the many trial cases that
12
need to be investigated to find a solution to wing rock. However, both experimental and
CFD investigations yield detailed flow characteristics that certaitlly could not be captured
with an inviscid model alone. The inviscid model developed will augment the number of
methods already being used. Inviscid modeling has been proven to capture a variety of
unsteady vortical flowfield characteristics while accomplishing the task within a couple of
hours on a mid size workstation.9 Primary investigation with an inviscid model would help
in limiting the experimental test matrix size by eliminating the methodologies that do not
demonstrate control of unsteady separated vortical flows within the limitations of the
model.
CHAPTER 2
LITERATURE SURVEY
2.1 Leading Edge Vortex Flow Control for Wing Rock
Study of leading edge vortex flows started in the 1950's on delta wing planfonn
designs. The advantages of using delta wings to generate leading edge vortex flows are
two told. Highly swept delta wings have favorable drag characteristics for supersonic
cruise at low angles of attack. Additionally, the leading edge vortex flows generate higher
lift at moderate angles of attack using strong vortices which emanate from the sharp
leading ,edges. A more balanced design can be achieved using delta wings since they have
both efficient supersonic cruise with capability to maneuver subsonically at high g
loading. 10
Control can enhance an aircraft's characteristics in the subsonic or supersonic
flight regimes. The leading edge vortex flow control can be applied to either supersonic
cruise or subsonic maneuverability, and for each of these regimes control can be
implement,ed statically or dynamically. As with many other types of aerodynamic research,
understanding control of static phenomenon historically has been pursued first since static
data is easier to obtain and is more intuitive. Static results often aid in understanding
dynamic phenomenon as well.
Studies in the early 1980's by Herbstll
, Lang and Francis12
, and Ashely13 showed
that a tactical advantage could be gained by exploiting increased subsonic maneuverability
13
14
in the high angle of attack post stall regimes of delta wings. This review will address only
those efforts aimed at increasing subsonic maneuverability either through static or dynamic
control of leading edge vortex flows. Several methods have been used recently to
investigate controlling separated leading edge vortex flows. These investigations faU into
three main categories: a mathematical controls only viewpoint, investigations that use
changes in geometry to control the separated leading edge vortical flows, and
investigations that manipulate the flowfield by blowing from the boundary layer.
2.1.1 Systems Approach
Hsu and Lan14 first developed a mathematical model of wing rock in 1985. Their
model was developed for one dimensional motion and three dimensional wing rock motion
to identify the major parameters involved in wing rock. The model was able to predict
wing rock closely using Beecham and Titchner's15 method to determine the parameters.
The assumed mathematical form for wing rock that they used is shown below.
~. =Lo+sinasLfJ ¢ +Lpo~ +sinasLpp I¢I~ +Lppl~l~
where ¢ is the roll angle, and as is the steady angle of attack.
Elzebda, Nayfeh and Mook16
,I7 compare Hsu and Lan's model and two other
derivative models to a slender delta wing mounted on a freetoroll sting. The derivative
models were developed by the authors to study the effect of the assumed nonlinear form
of the roll moment with roll angle and its derivative. They show that Hsu and Lan's
model cannot predict roll divergence since the original model only contains quadratic
terms. With the addition of a cubic term, the model predicts roll divergence and predicts
the wing rock motion more closely than the original model. Their second paper presents a
15
more global view of the wing rock phenomenon. The mathematical model was used to
construct phase planes which reveal the characteristics of wing rock such as stable limit
cycles, unstable foci, saddle points, and domains of initial conditions leading to oscillatory
motion and divergence. The phase planes of roll rate versus roll angle reveal primary
characteristics ofwing rock and the effects ofplanform geometry on those characteristics.
Luo and Lanl8 adopt Hsu and Lan's mathematical model for the nonlinear wing
rock motion. Luo and Lan only study the one dimensional case, however. The malO
objective of Luo and Lan's effort was to find a control function that would suppress the
wing rock motion. In order to do this, an arbitrary control variable, u was added to the
equation of motion.
if = Lo+ sinasLp t/J + Lpo~ + sin asLpp It/JI, + Lppl~l~ + u
Again, the resulting governing equation was solved using BeechamTitchner's
averaging technique which splits the solution into a inphase part for the frequency and an
outofphase part for the amplitude. The optimal control input to suppress wing rock was'
determined through a Hamiltonian method. The specific case for an 80 degree delta wing
was numerically solved using these methods to show that BeechamTitchener's technique
is accurate in analyzing dynamic motion and determining an optimal control input. For an
80degree delta wing, the following values for the aerodynamic characteristics were used :
Lo= 0.0
sinasLp =26.6667 sec2
Lpo =0.764785 sec1
sinasLpp = 2.92173 radsecl
16
Lpp =0.0
The most significant result that Luo and Lan obtained was that it is sufficient to use a
linear feedback of state variables, such as roll rate, to suppress wing rock. Also, the
system sensitivity to aerodynamic coefficients was determined to be a function of system
damping. Higher sensitivity was obtained for a system with lower damping.
2.1.2 Aerodynamic Control
Even though it was determined by Luo and Lanl8 that the optimal control scheme
was through the feedback of state variables, how to effectively affect those state variables
was not determined. Without understanding the aerodynamic mechanism involved in
separated leading edge vortex flows, any method that could influence the state variables
would be just as feasible as the next method. The second method controls separated
leading edge vortex flows through manipulations of the geometry. The major difference
between these methods and the controls only viewpoint is in understanding what is
happening aerodynamically. Although the controls viewpoint may actually use flaps or
another control device to stabilize the motion just as with the following efforts, the
controls viewpoint only use information available about the motion of the instability
through state variables. Control through geometric changes makes an effort to understand
the aerodynamic cause of the instability. The aerodynamic phenomenon driving the
instability is then controlled by altering the flowfield. The flowfield is altered by changing
the geometry in some manner which will be described below. These changes to the
geometry can be implemented passively or actively through feedback of a property of the
flowfield.
17
The first attempts at improving the capabilities of delta wings by modifying leading
edge vortex flows stemmed from the drag penalties associated with vortex lift. When
using vortex lift, subsonic maneuvering is potentially constrained by engine thrust or fuel
consumption due to drag. An alternative to this problem is to trade vortex lift for
potential lift by extending the angle of attack range of potential lift. 19 The range of
potential lift is extended by controlling the leading edge vortex flow. Potential flow is
achieved by promoting attached flow as long as possible. This is normally achieved
through leading edge bluntness. Robins and Carlson20 report that additional potential flow
achieved through leading edge bluntness is not counter productive to supersonic cruise
conditions, as it may seem, as long as the flow remains attached and the wings are swept
behind the Mach cone. In fact, supersonic cruise is enhanced by recovery of the leading
edge suction ofpotential flow.
The second reason for modifying or controlling leading edge vortex flows results
from stability problems in aircraft having swept delta wings. Control of longitudinal
stability problems as well as lateral stability problems such as wing rock need to be
controlled in order to gain more and more maneuverability.
2. 1.2.1 Geometric Control through Geometric Modifications
Rao16 reported in 1981 that for swept wings up to 40 degrees, leadingedge slots
and flaps have proven effective. However, during high g subsonic maneuvering, structures
which support the flaps are plagued by distortion effects which complicate actuation. For
delta wings swept over 40 degrees, Rao explores the use of fences, "pylon" vortex
generators, slots, and plates to ,enhance drag characteristics as well as control stability
18
problems passively. The four devices represented three fundamentally different
approaches: 1) modifications of leading edge upwash to obtain a camber effect with
fences and pylon vortex generators, 2) compartmenting the swept leading edge into "twodimensional"
segments using chordwise slots, and 3) forcing separation to produce a
leading edge vortex.
Using a 60 degree swept wing with fences, Rao found that optimum position for a
fence is located at the 50 % span point. Additional fences were found to be effective in
controlling spanwise flow and in promoting flow attachment. This increased the potential
lift and decreased drag. Drag reductions up 25% were found when using pylon vortex
generators. By delaying separation, these methods delayed the onset of longitudinal
instability also.
Slots were found to reduce drag as the other fence methods although the
characteristics of the device were different. Dynamic longitudinal instabilities were
aggravated, however. Vortex plates were able to reduce drag and alleviate the
longitudinal instabilities by introducing a component of thrust along the leading edge of
the delta wing.
Mar,chman21
,22 conducted wind tunnel test to determine the aerodynamic effects of
leading edge flaps deflected upward. Marchman used a 60 degree and 75 degree swept
wing. Various sizes and shapes of leading edge vortex flaps were used. It was found that
inverted vortex flaps created strong lift, but the flaps also created additional drag. Large
changes in lift were not accompanied by large changes in pitching moment. With these
characteristics, a properly designed negatively deflected flap may be desirable for landing
19
conditions. Leading edge vortex flaps were also found to be more effective as sweep
increases.
Grantz and Marchman23 study the effect of trailing edge flap deployment on leading edge
vortex flap aerodynamics. On the same 60 degree and 75 degree delta wings as above,
they establish that trailing edge flaps do not significantly improve the vortex flowfield.
Comparisons of computations generated to study effects of yaw and vortex flaps were
made by Murman24 and Powell and Murman25 in 1986. The computations were generated
and validated for the conical Euler Equations in the supersonic flight regime for an ideal
flat plat,e delta wing whose geometries included thickness, sharp leading edges, and two
vortex flaps. Munnan and Rizzi26 review the applications of Euler equations to sharp edge
delta wings with leading edge vo,rtices. Freestream Mach numbers from zero to
supersonic are discussed.
Ng and Malcom21 investigated how the forebody vortices on a F/AI8 could be
controlled. The flowfield ofa highly slender forebody at high angle of attack is dominated
by vortices which can present stability problems if they become asymmetric. Ng and
Malcolm implement a small rotatable strake on the forebody which can be fixed in place or
deployed actively. The strakes generate vortices from the leading edge and trailing edge.
These new vortices dominate the forebody flowfleld and help maintain attached flow on
the forebody. More importantly, the flow asymmetries are able to be controlled and
exploited. The strakes could generate yawing moments of different magnitude by moving
the strakes to different angular positions. This method of controlling separated vortices
was shown to be highly effective in controlling the flow on the forebody over a wide range
20
of angles of attack and sideslip. The strength of the vortices generated by the strake can
be manipulated by changing the shape of the trailing edge ofthe strake.
Synolakis, et at.28 use extended leading edge winglets to study passively controlling
ddta wing rock. The winglets are flaps which linearly increase in chord on the leading
edge ofthe wing. Other authors such as Klute31 have termed similar types of flaps as apex
flaps. The basis for using the winglets is to interfere with the fonnation of the primary
vortices emanating from the leading edge. They summarize that the critical angle at which
the onset of wing rock occurs is highly dependent upon the particular geometric
configuration. Delta wing geometries with winglets behave much like the baseline delta
wing except that the onset is delayed. Extended winglets with a wingletIength to chordlength
ratio of 0.43 appear to increase the wing rock envelope the most. Based on their
result for an 80 degree delta wing, the angles of attack where the onset of wing rock
occurs can be delayed by as much as 25 degrees for moderate angles of attack. By
delaying the onset, passive control can be an effective method for improving the stability
of the delta wing by expanding performance envelope. No flow visualization was done to
confirm the objective ofinterfering with the fonnation of the primary vortex structure.
Walton and Katz29 used control flaps on the leading edges of a onedegreeof
freedom in roll model. By driving the flaps out of phase with the roll angle, wing rock
was suppressed. The amplitudes of the flap oscillations were such that primary vortices
were modified just enough to suppress wing rock. Walton and Katz state that the flaps in
their investigation would be more effective if the leading edge of the wing was thinner.
Also, the flaps would be more effective if the placement of the flaps was farther forward.
This indicates that the control mechanism used modifies the characteristics of the
21
separation and the rollup. They conclude that the same method theoretically could be
implemented along with an active control scheme to suppress wing rock on an actual
aircraft configuration.
For a delta wing in a slight side slip, Ng, Skaff, and Kountz30 developed a
methodology using flow dividers on the top of the wing to control wing rock. Flow
dividers are vertical fences mounted on the upper side of the wing. The fences or dividers
attempt to decouple the flow of each side from the other side's effects. Their results
provide an interesting insight into the phenomenon of wing rock itself By decoupling the
flowfield, it was hypothesiz.ed that ron oscillations would diminish. Ng, et aI. investigated
the effects of divider geometry, sizes, and placement and the effect upon the roll
oscillations. They were able the suppress wing rock for a wide range of angle of attack;
however, at the lower range of a's where wing rock first occurs, the divider actually
amplified wing rock. As expected, the divider decreased the roll moment for moderate
angles of attack, damping the wing rock oscillation. Also, for angles of attack where the
divider amplifies the wing rock oscillation, the divider increased the roll moment at a side
slip condition. The divider was able to damp the wing rock motion because it decreased
the vortical interactions taking place in the flow field. Also, damping occurred at higher
angles of attack where vortex breakdown was present partly because the breakdown is
asymmetric and out of phase with roll rate. The asymmetry in the position of the vortices
is due to the vortices dependence upon the flow conditions at separation.
Klute, et aI. 31 performed an experimental study on controlling vortex breakdown
on delta wings. Several control surfaces were tested in fixed and dynamically pitching
delta wings. Klute used flow visualizations, surface pressure measurements and Laser
22
Doppler Velocimetry measurements to map the flowfield. Vortex breakdown was delayed
by implementing a drooping apex flap or "winglet". An apex flap is a leading edge flap
whose chord increases linearly. An apex flap typically has a chord which is a set
percentage of the local s,emi span of the wing. Delays by as much as 8 degrees past the
steady state angle of attack for breakdown were achieved.
Syverud, etal.32 implemented an 80 degree leading edge extension as a vortex flap
on a 70 degree delta wing. The results obtained were based on qualitative flow
visualization studies performed in the 16 x 24 inch water tunnel at NASA  Langley. The·
leading edge extension is body hinged and serves to control the roll oscillations. The joint
between the trailing edge of the leading edge extension and the leading edge of the main
wing is swept forward. Dihedral leading edge extension deflection was found to stabilize
the primary vortex system whiIe it was found that an anhedral deflection destabilized the
vortex system. Once destabilized, the vortex system exhibited rapid vortex breakdown.
The delta wing has three vortex structures which dominate the flowfield: the leading edge
extension vortex, inboard wing vortex, and outboard wing vortex. With no deflection of
the leading edge extension (LEX), the flowfield is similar to delta wing flowfields reported
by other investigators. As angle of attack is increased vortex breakdown advances
upstream, and it appears on top of the wing at the trailing edge at an angle of attack of 33
degrees. With no LEX deflection, no inboard wing vortex is present. As the LEX is
deflected upward, the flowfield stabilizes. The upward deflection strengthens the
outboard vortex by displacing the wake disturbance from the LEX. Downward deflection
of the LEX causes breakdown of LEX vortices to move farther up on the wing while
preventing the outboard vortices from forming. Vortex positions change with LEX
23
deflections also. As an upward or dihedral deflection increases, the LEX vortex is bent
outboard and clos,er to the wing by the outboard vortex. The bevel of the leading edges
and trailing edges of the delta wing were found to destabilize the vort,ex system if the
bevels were on the upper surface of the wing. The upward facing bevels also bend the
LEX vortices outboard.
Rinoie and StollerY3 investigated the use of vortex flaps and vortex plates on delta
wings as a means to improve the lift/drag ratio. Force measurements were obtained along
with surface pressure measurements in a low speed wind tunnel for a 1.15m span, 60
degree delta wing. The tunnel speed was set at 30 mls. Results for the vortex flap
showed that the lift/drag ratio improved with the deflection of the vortex flap. The
improvement was attributed to the flow either not separating at the leading edge of the
vortex or the flow reattaching to the flap after separating. The normally large separated
zone would be small andencIosed. By keeping the flow attached, the lift increased. The
tests were performed by deflecting the flap from 0 to 60 degrees as angle of attack is
swept from 8 to 57 degrees. The results for a flap deflection of 30 degrees are presented
due to the 30 degree deflection showing the best performance over a wide CL range. They
demonstrate that as far as improving the lift/drag ratio is concerned no improvement is
seen once the flow is completely separated at angle of attack of about 35 degrees. Before
30 degrees angle of attack, the greatest improvement in LID ratio for a leading edge
vortex flap deflection of 30 degrees is 40% at a CL of 0.45. AU pressure measurements
were performed at x/er = 0.4 and x/er= 0.8. By using the pressure measurement data, the
effect ofthe vortex flaps on the pitching moment, em, is little if not at alL

24
The spanwise pressure distribution reflected the reattachment of the flow at low
angles of attack up to about 6 degrees for no flap deflection. A definite formation of the
primary vortex structure typically found on delta wings occurs between ex = 12.4 and 37.0
degrees. By a. = 37 degrees, the pressure distribution spreads out signifying the arrival of
vortex breakdown on the upper wing surface. The vortex flow completely collapses at a.
= 48.7 degrees as demonstrated by the pressure distribution being flat. Much the same
steady results are obtained for the 30 degree flap deflection case except that the flow
remains attached until about an angle of attack of 12 degrees. So, vortex flaps are able to
increase the LID ratio by increasing lift and by reducing drag by reattaching the flow at
lower angles of attack. The shape of the low pressure peaks due to the primary vortex
structure is slightly different between the two flap setting suggesting the location and/or
strength changes with the flap setting. This is plausible since the vortex flap is able to
reattach flow at lower angles of attack. Rionoie and StoUeryH present pressure
distribution results at x/er=O.4 where the flap angle is swept from 0 to 60 degrees while
angle of attack is held constant at 6 and 12 degrees. At an angle of attack of 12 degrees
for flap angles up to 30 degrees the primary vortices move outboard as the flap angle
increases. As the flap angle is increased past 30 degrees the separation at the leading edge
of the flap attaches and a line of separation forms just inside ofthe hinge line.
The effect of vortex plates are also investigated for improving the LID ratio. As
implemented in their study, a vortex plate is a thin plate fastened to the lower side of the
leading edge. The plate serves to extend the separation point out from the actual wing's
leading edge. The vortex plates were able to obtain result comparable to that of the 30
degree flap deflection. The best results for the vortex plate are seen when the vortex plate
25
protrudes ahead of the leading edge ofthe wing. For the vortex plate to be effective, the
amount that the plate protrudes from the leading edge ofthe wing needs to be constant. If
the protruded amount is tapered and increasing from the apex, no improvements in the
LID ratio can be ascertained.
2.1.2.2 Pneumatic Control
Again, the investigation of control usmg blowing was two fold. Control
investigations started by investigating drag reduction and later applied drag reducing
control techniques to stability issues.
The aerodynamic effect associated with blowing a jet spanwise over a wing's
upper surface in a direction parallel to the leading edge was investigated by Campbell34
For delta wings, arrow wings, and diamond wings with sweep angles of 30 degrees and 45
degrees, spanwise blowing was shown to aid in the formation and control of the leading
edge vortices. Campbell demonstrates that blowing rates must increase with spanwise
position in order to achieve fun vortex lift at a particular spanwise station. The effects of
blowing are the generation of larger increases in lift at high angles of attack, improvement
of drag polars, and extension of the linear range of pitching moment to higher lift
coefficients Lifting efficiency of the spanwise blowing is judged by detennining the lift
augmentation ratio, ~CJC~. This is shown versus angle of attack in degrees in the next
figure. As angle of attack increases, the lift effect of the spanwise blowing becomes
greater than the effect of the blowing jet thrust acting vertically at about an angle of attack
of 15 degrees. The largest augmentation ratio was obtained at the lowest values of
blowing rates. As blowing rates increased the lifting efficiency of the jet decreased.
26
Smaller amounts of blowing increase the jet induced camber effect on the wing, and thus
the overall lift at angles ofattack above 15  16 degree is increased.
Celik, Roberts, and Wood35 investigate the ability of tangential leading edge
blowing to stabilize and control flow asymmetries on a delta wing at high angles of attack.
Their experimental effort also investigated the ability to control flow instabilities in the
flowfield such as vortex breakdown. For various pitch, roll, and yaw configurations,
steady state force, moment, and pressure distribution data was obtained. Pitch
configurations included post stall settings. The delta wing model used has a 60 degree
leading edge sweep angle. Tangential blowing on the leading edge is implemented by a
linearly varying slot which extends from the apex for the entire leading edge. Results
obtained indicate that vortical flow can be controlled up to high angles of attack. The roll
moment reversed for poststall angles of attack when compared to the prestall
measurements. Control reversal due to the reversal of roll moment was diminished by
decreasing the effective angle of attack with symmetric blowing. Blowing is found to be
effective at different roll and yaw configurations. Asymmetric condition for prestall
angles of attack can be created by superposing force and moment conditions.
Wong36 used leading edge tangential blowing with an active control scheme. The
blowing scheme used asymmetric blowing from rounded leading edges. By using a
symmetric blowing configuration, the wing rock oscillation amplitude was reduced. But,
by using the asymmetric configuration and active feedback control, wing rock was damped
considerably within one cycle of the limit cycle motion. The success of trus technique to
modify the wing rock motion is due to the ability of the blowing to control the position of
the separation points.
27
Suarez, et at. 37 conducted experimental freetoroll tests on a 78 degree swept
delta wing. The suppression of wing rock was investigated by using forebody blowing
where the forebody is the area on the body preceding the main wing. This study
specifically uses the nose area of the model as opposed to the body area ahead of the
wing. Several blowing techniques were investigated as a means of suppression. Blowing
tangentially aft from side nozzles on the forebody was shown to damp the roll motion at
low blowing rates and stop it completely at higher blowing rates. The higher blowing
rates created flow asymmetries whiIe eliminating the dynamic hysteresis in the vortices'
strength and position. The steady flow asynunetries caused the wing to stop at nonzero
roll angles.
The second technique used forward blowing and alternating left  right pulsating
blowing This technique was more efficient and could damp the oscillation almost
completely at lower blowing coefficients than in the tangential blowing technique. No
major vortex asymmetries are induced at lower blowing coefficients.
Greenwell and Wood38 use tangential leading edge blowing to demonstrate roll
moment reversal as reported by Celik, Roberts, and Wood35
. Similarity ofthe roll moment
reversal to resuhs obtained for high sideslip angles was reported. Blowing on a single
leading edge was demonstrated to reduce the effective angle of attack for the blown side
while adding an effective sideslip angle. The decrease in effective angle of attack on the
blown side yielded increases in the leading edge normal force resulting in a net "blown
wing up" rolling moment.
By rounding the leading edges of the wing, blowing air tangent to the surface
energized the boundary layer. The jet transfers momentum to the outer flow which delays
28
the separation ofthe outer flow itself. By providing control of the separation location, the
location and strength of the primary vortex core and its associated feeding sheet can be
modified. Also, blowing keeps the vortex system closer to the wing since the effective
angle of attack is reduced. The vortex system ES able to generate more lift on the blown
side ofthe wing resulting in the blown wing rotating upward.
Bean, Greenwell, and Wood39 apply tangential leading edge blowing to problems
which occur in buffeting of fins or airfoils. Two 60 degree sweep delta wings were used
to study buffeting. One wing or fin was rigid and instrumented so that surface pressure
data could be obtained. The other wing was flexible to study buffeting response of the
wing. The experimentation showed that the buffeting pressure profiles and the response
of the delta wing matched each other very closely. Qualitatively, the "effective angle of
attack" of the primary vortices was reduced when symmetric leadingedge blowing was
used. Symmetric blowing at a constant rate shifted the buffet excitation and response to
higher angles of attack. Flow visualization of the flowfield confinned that the fluid
buffeting mechanism was only shifted and not eliminated. With the use of an optimum
blowing configuration for each new angle of attack, the response to the buffeting has
potential to be completely suppressed.
Crowther and Wood40 experimentally investigate yaw control through tangential
forebody flowing. Foroe and moment data was measured for angles of attack up to 90°
for a number of different slot geometries and locations. It was found that small blowing
rates from short slots at the front of the forebody provided iargercontrolled yawing
moments at about a = 60°. Using larger slots and larger blowing rates was shown to
provide some degree of control up to 90 degrees. Control was demonstrated despite the
29
loss of coherent vortical flow structures at higher angles of attack. Crowther and Wood
identify that yawing moments due to blowing are dependent upon geometry of the body
and the slot. Thus for some geometries flowfield coupling between the forebody and wing
can lead to unexpected yawing moments and roll moment excursions.
More succinctly, at higher angles of attack, blowing on the left: side yields nosetoleft
yawing moments and blowing on the right side yields nosetoright yawing moments.
As angle of attack increases throughout the range studied rudder yaw control power
decreased while yawing moment available from blowing increases for a given blowing
rate.
Discrepancies in the definition of the blowing coefficient showed dependencies of
several different parameters. For constant area slots and incompressible flows all
definitions are the same, but for different slot areas CJl describes the trends much more
effectively. CJl is defined as:
Piet .Ajet .(Viet)2
CJl ==''
q·S
Control reversals that occur at low blowing rates are associated with the expected
tangential forebody blowing fluid mechanisms reversing. This reversal can be minimized
by careful forebody/slot geometry design. Crowther and Wood note that jet massflow
requirements are within reasonable engine bleed flow levels. Kramer et a1. 41 report much
the same findings as Crowther and Wood.
"
30
Celik, Pedreiro, and Roberts42 study several forebody blowing schemes to aid in
eliminating wing rock. They use tangential forebody blowing to provlide lateral control
through forebody/wing interactions on a sharp leading edged modeL The usefulness of
active control through blowing schemes including symmetric, asymmetric, steady, and
unsteady blowing was demonstrated. Experimentation showed that the wing rock motion
could be suppressed by the steady, symmetric or unsteady, asymmetric tangential forebody
blowing. .Differences between the symmetric and asymmetric cases were found where
asymmetric blowing was found to be very efFective.
Ng et al.43 independently corroborate the findings made by Celik, Pedreiro and
Roberts. Tests were conducted on a slender forebody on a 78 degree swept delta wing in
a water tunnel. Steady blowing tangential from nozzles at the tip of the forebody was
found to be capable of suppressing wing rock. At low blowing rates the motion was
attenuated while at high blowing rates the motion was eliminated. Due to the higher
blowing rates inducing vortex asymmetries on a time averaged basis, alternating pulsed or
unsteady blowing on the left and right sides of the forebody was shown to be effective in
suppressing wing rock without creating timeaveraged flow asymmetries.
Wong et al. 44 demonstrated experimentally with a freetoroll wind tunnel model
that significant rolling moments could be produced up to an angle of attack of 55 degrees
using tangential leading edge blowing. A pair of fastcontrol servo valves were designed
and built to implement an automatic feedback roll control algorithm on a digital controller.
Results show that wing rock could be damped at angles of attack up to 55 degrees. By
modifying the control algorithm to use asymmetric blowing, wing rock was eliminated in
less than one cycle of the limit cycle osciHation. Control reversal between prestall and
31
poststall angles of attack found previously by other investigators was eliminated using
asymmetrie blowing with eontroL Roll eornmand following was shown to be attainable
with the use of a feedforward gainscheduling control algorithm.
Arena, Nelson, and Schiff5 investigate problems in directional control of aircraft
through pneumatic blowing. An aircraft model with a chiDed forebody was used to
quantify the effectiveness of blowing through a slot in the chine to affect the aircraft's
lateral stability. Comparisons to the baseline planfonn are mode with control deflections
of the rudder and with the tail on and off. Results were collected for several blowing
coefficient within an angle ofattack range from 0° to 75°. Through flow visualization as
well as force and moment balanc,e data, results obtained revealed several facts. First, a
conventional tail configuration on an aircraft loses it effectiveness at higher angles of
attack. In this region, blowing was demonstrated to be an effective altemativ,e since the
moment generated by blowing was four times greater at the maximum than the moment of
the jet's momentum. Thus, the interaction of the blowing jet with forebody flow is
important in generating the large forces and moments on the aircraft. As angle of attack
increases, the moments and forces due to blowing increase until the angle of attack forces
the blowing fluid mechanism from interaction with the forebody flow of the aircraft. For
the planform studied the loss of blowing moment occurred at 60°approximately. The
model used was a chined forebody on a 50° swept diamond wing with a single vertical tail.
The tail control study revealed that strong interactions exist between the forebody
flowfield, the separated wing flow, and the vertical tail. The control authority of the tail is
unique for any type of tail configuration. The merit of each tail configuration would have
to be investigated before it is implemented in a design. For the specific geometry stated in
.,'\
r
, "
32
certain blowing cases, the lateral controllability was markedly improved. With the
conventional rudder along with blowing from the chines, the iateral controUability was
almost doubled.
Gittner and Chokani46 hypothesize that effects of the nozzle exit geometry are
important in forebody vortex control when blowing is used. Moksovitz et aI. showed the
effect ofdecreasing the vortex asymmetries on the forebody. Therefore, since vortex flow
characteristics could be changed by small surface perturbations, nozzle exit geometry may
yield additional control ofvortical flows. The experimental effort by Gittner and Chokani
show that both height and width ofthe blowing nozzle exit geometry was important. The
most effective geometry of these was a low broad nozzle blowing aft along the forebody.
Gittner also concluded that the degree of synunetry in the vortical flow before blowing
actually moderates the blowing effectiveness.
A survey of past work has shown three methods studied for control of leading
edge vortical flows for wing rock. The first method uses a systems approach to quantify
the wing rock roll oscillations, identify controllable parameters, and then mathematically
arrive at a control algorithm. The most significant result from this type of study revealed
that linear feedback of state variables such as roll rate would be sufficient for control of
wing rock oscillations. The remaining methods can be categorized as aerodynamic
methods where a further understanding of aerodynamics of the flowfield was applied.
Within these aerodynamic methods, the two methods of control modify the flowfield
through geometric modifications of the planfonn or through pneumatic methods.
Geometric modifications of the planfonn through the use of vortex flaps, vortex plate,
apex flaps, vortex generators, and aerodynamic fences were studied. These methods
"I'
,f
I·
.II' .
:1
33
prove effective in modifying the location of the separation point of the primary vortices as
well as the position and strength ofthe primary vortices themselves.
Blowing methods imp,lement,ed tangential leading edge blowing and forebody
blowing. Blowing was found to be more successful in controlling leading edge vortex
flows when moderate flow rates were used to modify the location of the separation point
of the primary vortices and the position and strength of the primary vortices. Higher
blowing rates were demonstrated to disturb the flowfield enough to eliminate the
effectiveness ofthe separated flow or in some cases the separated flow itself
The combination of the three methods show that a control scheme that has the
ability to modify the separation point, position, and/or strength ofthe primary vortices can
potentially be effective in controlling leading edge vortical flow found in wing rock
oscillations. Control using linear feedback of state variables is sufficient and can be used to
determine optimized control algorithms.
,
( .
j, '
II :,
CHAPTER 3
METHODOLOGY
3.1 Survey ofInviscid, Potential Flow Modeling
A background study of modeling of separated vortical flows is useful in
understanding previous successes and limitations of using inviscid, potential flow
assumptions. Each type of vortical flow is unique in several aspects; therefore, a
discussion of separated vortical modeling only as it relates to sharp leading delta wings
will be reviewed. The uniqueness of this flow field is characterized predominantly by the
flow separating from the sharp leading edges with the shear layer of the separation
coalescing into two primary vortices.
The earliest investigators, such as Legendre47
, Brown and Michael4
, have used
potential vortex models to represent steady delta wing flow fields. They found that sharp
leading edge delta wings inherently have leading edge separated vortical flow fields. The
separation of the flow is caused primarily by viscosity in the boundary layer fixing the
separation point at the J,eading edges. These separated shear layers coalesce into two
primary vortices on the suction side of the delta wing. Visser48 later confirmed through
hot wire anemometry that the majority of the vorticity in the flow field is concentrated
into the two leading edge vortices. The location of the vorticity on a steady delta wing
34
35
flow fie~d was determined through cakulation of the circu~ation by perfonning a line
integrID around the velocity field data. It was shown that the majority of axial vorticity
found in these leading edge vortices is found in the viscous core region of the vortex.
The diameter of the core region before vortex burst is on the order of about 5% of the
Jocal semispan. Since the vorticity is concentrated into two regions that are small when
compared to the span of the wing, the flow field can be assumed to be inviscid as ~ong as
vortex breakdown does not occur. Thus, the two concentrations of vorticity in the
primary vortices can be modeled using a potential vortex with a viscous vortex core.
Additionally, if the vorticity present at the leading edges is modeled macroscopically, the
entire flowfield can be modeled by inviscid fluid mechanics. Consequently, these
assumptions are very feasible where vorticity is either neglectible or where it is
concentrated. If the vorticity is concentrated and not neglectible then the effects of the
vorticity can be modeled macroscopically. For this study, this assumption is good at
angles of attack where vortex breakdown is not present on the wing.
In more recent rnodels49
•
50
,4.5 several additional theories have been used to further
simplify the flowfield model. Typically, slender wing theory has been implemented along
with a conical flowfield assumption. Slender wing simplifications are implemented by
neglecting the gradients of the flowfield in the axial direction when compared to those in
the cross flow plane. The slender wing theory is justified since the length ofthe delta wing
is much larger than the thickness.
By using a conical flow assumption, the properties of the flowfield are assumed to
be invariant along rays emanating from the apex of the delta wing. The root chord and
36
local semi span of the wing scale ithe dimensions of the wing and its aerodynamic
characteristics so that the aerodynamic properties are function ofy/x and zJx only.
Several conditions must be placed on the use of the conical assumption in a delta
wing flowfield. This assumption can be justified experimentally over the forward part of
the wing for steady flows. However, as the trailing edge is approached, the conical
assumption fails due to trailing edge asymmetries and other effects. Under unsteady
conditions, the conical assumption fails for the forward part of the wing because of the
unsteady boundary condition. For roll motion the rigid body velocity introduced into the'
boundary condition is not conical since the rigid body normal velocities vary with the
linear distanoe from the axis of rotation.
The justification for using the conical assumption is provided by Arena50
. Arena
states that due to these reasons discussed above the unsteady flow is not "globally"
conical. In experimental efforts by Arena and others, the local flow does not strictly
follow conical rays. However, the pressure distributions, vortex positions, and roll
moments are qualitatively selfsimilar. The consequence of only qualitative self similarity
is that the flowfield can be solved at any crossflow plane, but the results depend on the
chordwise position of that crossflow plane. This dependency is a scale factor. Therefore,
the properties have scaled similarity. The scaled factor introduced by the conical
assumption will be detailed in the next section.
As can be seen in the following figures, the static position of the vortex cores is
three dimensional. As chord station increases, the vortices move to higher and higher
positions above the wing. Likewise the vortices move out from the centerline ofthe wing.
37
Compom:Dl
of,elocCly
:c~
V.J:(lSO:... ~
Side View
lollt~ Vortex SOOWI')
,
,. ' '
CumPoRCN
of ",cIociI)'
:JCf'OSS lICnc)I;
Top View
Three Dimensional Trajectories ofthe Primary Vortices (Arena)
Figure 3.1
The conical assumption is very important because it reduces the solution in three
dimensions to a scaled single chord station when coupled to the slender wing assumption.
Subsequently, the model is self similar in the axial direction with the similarity being a
function of the local chord position and semispan. After applying the conical assumption,
the model is referred to as quasi three dimensional and not two dimensional because the
axial component of velocity is still very much a factor in the model. If the axial
component ofvelocity was not present in the solution, the primary vortex positions would
convect normal to the wing surface instead of parallel to the free stream.
The solution of the model thus far would be defined if the mathematical solution
were unique_ However in addition to the physical problem, the potential of the model thus
38
far is not unique or more precisely, it is not mathematically single valued. Different
investigators have approached this problem from various directions. The major technique
is fonnulated by Brown and Michael4
, and refined by Mangler and Smith5
. The difference
between these techniques is subtle. By observing that a simply connected region would
provide a unique solution, a mathematical branch cut emanating from the leading edge and
ending at the center of the primary vortex will make the potential of the model single
valued. Additionally, a physical mechanism is needed to conduct vorticity from the wing
where it is generated to the core of the primary vortices. Brown and Michael model this
physical mechanism or "feeding sheet" with a straight branch cut that stretches from the
leading edge to the center of the primary vortex on each side. This solves the single
valued problem while also providing a model for the physical shear layer coming off the
leading edge. A noted liability of using a straight branch cut is that the gradient of the
potential is zero across the infinitesimal line that represents the feeding sheet. Since the
gradient ofpotential is zero on either side of the branch cut, the branch cut can sustain no
load. Tills is true of physical shear layers. However, shear layers in this particular
situation are physically not straight. They always follow streamlines of the flow. If the
mathematical branch cut does not coincide with a shear layer on a streamline, then
additional error is introduced into the model. Mangler and Smith recognized this liability
and use the same technique except that their feeding sheet is curved and follows the
stagnation streamline from the leading edge to the center of the primary vortex. This
second technique is more accurate in that a force cannot exist across a shear layer or a
streamline. Using Brown and Michael's linear model, a force would exist on the branch
cut.
39
a) actual fI.owfield
x
b) appmximated fiewfield
Assumed and Approximated Ftowfield Sketches
(Brown and Michael 1955)
Figure 3.2
40
The model presented here uses the linear branch cut for simplicity in its
implementation since the curved branch cut added additional complexity to the model and
yielded little in terms of accuracy. When using a branch cut that does not follow a
streamline, the position of the feeding sheet must be found by iterating the flowfield
solution until no force exists across each feeding sheet. From the crossflow plane
vortices either. In tlUs effort, the model enforces the no force condition on the branch cut
with the force balance can be seen in Figure 3.3.
+s y
rr
~I
Fvr
Fsr 7
z
s
~
Fvl \
~FSI
n
and the primary vortices together for each side. With the specification of the no force
conditions, the steady state model of a separa.ted vortical flowfield is complete, and the
detenruned by the using the ,conical assumption, no forces can exist on the primary
solution will be mathematically unique. The model showing the linear branch cuts along
Vinf· Sina.
Delta Wing Crossflow Plane with Branch Cut
Figure 3.3
41
Arena and Nelsonso extended the use of the potential model with linear feeding
sheets by applying it to the unsteady flowfield. Solving a dynamically changing flowfield
is more complicated than solving a steady state flowfield at discrete time intervals. Much
ofthe extension ofthe potential model to the unsteady flowfield case is not intuitive. For
example, the potential of the flowfield at each point is not constant as it is in the steady
case. The potential of the flowfield varies with time at each location. Despite additionaJ
complications, Arena and Nelson have shown that the essential characteristics of the
unst,eady delta wing still can be captured by modeling only the primary flow
characteristics, namely the primary vortices. The model developed by Arena and Nelson
used a conformal mapping technique to solve the flowfield. The conformal mapping can
provide an exact solution £or a given model; however, the thickness of the wing was not
included due to the complexity oftransforming the "thick" model into the complex plane.
The solution to the present model will be obtained by using a panel technique
where the body geometry represented by a distribution of constant strength sources and
vortices. The source strengths for each panel will vary while the vortex strengths will be
constant and equal for all panels. This solution method will allow the delta wing to have
an arbitrary thickness distribution allowing investigation of variations in planform
geometries.
42
3.2 Solution ofMathematical Model
The mathematical representation and solution of the unsteady delta wing model is
detailed in the following three categories:
1. Definition of Governing Equations
2. Governing Equations Solution Method
3. Extension ofModel to Unsteady Motion
3.2.1 Definition of Governing Equations
The model is mathematIcally defined by stating conservation of mass along with
simplifications such as incompressibility, irrotationality, and wing slenderness as it applies
to the delta wing flowfield.
3.2.1.1 Unsteady Conservation ofMass
The major assumptions of the model are that the flowfield is incompressible,
inviscid, and unsteady. The continuity equation reduces to yield the governing equation as
follows. The continuity ,equation is
v·q=O
Since the flowfield is irrotational,
where q= (u, v, w) (Eq. 1)
Vxq=O
A potential function can be defined such that
at>
u= a and
at>
v= q; and
at>
w=
&
43
Substituting the potential function definition for the velocity components into equation 1,
The governing equation of the flowfield is then
(Eq.2)
3.2.1.2 Slender Wing Assumption
The next assumption made in developing the model is that the wing is a slender
wmg. The slender wing assumption states that gradients with respect to the x direction
are negligible compare to the y and z directions in the crossflow plane. In other words,
the crossflow plane is dominant.
o«00
Ox iJy' &
3.2.1.3 Conical Assumption
A dominant crossflow coupled with a conical flowfield assumption anows the
crossflow planes of the delta wing to have selfsimilarity. With these two assumptions, the
governing equation reduces to two dimensions in the crossflow plane. Note however that
the solution is still quasithree dimensions since the axial component of velocity is still
present.
!
I'
"'I
:l
I~ ii
:~ ,
1=1' i~' ,.,1
:~I :::)
(Eq.3)
......
44
In Figure 3. 1, the geometry of the wing is defined as having a local semispan, s, and a
sweep angle, A. Furthmore, E = 90°  A. For conical variables anywhere on the wing, the
chord position is scaled by the local semispan.
• x
x ==tane
s
The crossflow dimensions are nondimensionalized by applying the corneal assumption.
y* = yls and z* = zls
3.2.1.4 Boundary Conditions
To obtain the solution to the 2D Laplace's equation for flow on a body, boundary
conditions must be stipulated on the boundary itself and in the far field at infinity. The
boundary condition on the boundary is such that the flow must be tangent to the surface of
the body. The far field boundary condition states that the perturbation velocity in the flow
field must be zero at infinity. Stated another way, the total potential of the flowfield at
infinity must be equal to the potential ofthe freestream itself.
On the surface ofthe body,
V<'P· ii = Vn
where n is a unit vector normal to the surface of the body and Vn si the nonnal velocity of
the rigid body on the surface ofthe wing. For steady conditions, Vn = o.
45
In the far field at infinity,
V<I>=q"" sin a
r~""
3.2.2 F10wfie1d Solution
The solution of the conical, slender wmg Laplace's equation is obtained by
superposing potential singularity solutions in the crossflow plane along with laws
governing vorticity generation and shedding.
3.2.2.1 Singularity Solution Superposition
The singularity solutions are distributed along the surface of the wing itself by
discretizing the wing geometry into a number of linear panels that approximate the actual
shape. The wing is discretized using cosine spacing to yield better resolution of the
flowfiefd on the geometry at the leading edges. Cosine spacing allocates the panel end
points according to a constant angle of li~ = 2n/m where m is the number of panels. The
independent coordinate of the end points is described by
c
x =(1 cosP)
2
while the dependent coordinate is determined by the desired geometry. By distributing a
singularity solution on each panel, the flowfield about any arbitrary shape can be found.
Each panel also has a collocation point defined at its center. The collocation point
is the geometric location where the boundary conditions are enforced. For this study, the
flow tangency boundary condition for each panel will be satisfied at the collocation point
of each panel by requiring the sum of the velocities generated by all of the singularity
, ,
46
solutions to be equal to the Donnal velocity of the wing. The boundary condition at
infinity states that the perturbation ve10cities induced by the wing must be zero. This
boundary condition wil1 be satisfied by choosing singularity solutions that automatically
satisfY the farfield boundary condition.
This method is known as a panel method. In contrast to the conformal mapping
technique, the panel method technique allows a solution to be obtained that can account
for thickness of the geometry. However, the panel method is an inexact solution to an
inexact model whereas confonnal mapping is an exact solution to an inexact model.
Therefore, for a thick wing, a tradeoff exists between complexity of the transform in
complex variables and the errors introduced by using a panel method.
A number of combinations of singularity solutions work in this application. So, the
choice ofthe actual singularity solutions to be used is somewhat arbitrary. For this model,
constant strength source and vortex distributed singularity solutions are used on the
boundary to approximate the geometry of the delta wing in the freestream flow. The
equation for the velocity at a point due to a constant strength distributed source is shown
below along with the velocity due to a constant strength distributed vortex. The
derivation ofthese equations can be found in Katz and Plotkin51
After applying the conical assumption, the velocity due to a constant strength distributed
source in local panel coordinates is given by the following equations:
47
[(
.. *)2 ..2]
U * = (J* ln Y  Yl + Z
p 47r (" ")2 .2 Y  Y 2 +Z
. [" ..] '. _ (J . J Z I Z
W p  In tan (.. * )  tan (. .)
27r Y  Y2 Y  Y.
where
Y"=Ys
" Z Z =;
s
r"= r Va> sin a
u " =(j 
Va> sin a
r"= __r_
Va> sin a
The velocity due to constant strength distributed vortex in local panel coordinates is:
where (y,z) are the coordinates of the collocation points, (yI,ZI) are the coordinates ofthe
counterclockwise most end, and (Y2,Z2) are the coordinates of the clockwise most end.
48
The velocities due to the free vortices used to model the primary vortices are
The cent,er of the potential vortex origin is located at point (Yo,zo) while the point of
interest is located at the point (Y,z).
The arbitrarily chosen singularities of a distributed constant source and a
distributed potential vortex are solutions of Laplace's governing equation individually.
Using superposition, a linear combination of both of the individual solutions can be found
such that a particular set of boundary conditions is satisfied. Since these singularity
solutions also automatically satisfy the far field boundary condition, the remaining
conditions to be satisfied are the boundary condition of flow tangency on the wing along
with specification of circulation in the flowfield. The boundary condition on the surface of
the wing is enforced at the collocation point of each panel.
The solution scheme depends upon whether the equations govermng the
circulation in the fIowfield are linear or nonlinear. The equations specifying the flow
tangency on the panels are linear. As a result, all of the flow tangency equations along
with the linearized circulation equations can potentially be placed in a matrix £onn and
49
solved by LV decomposition, a method of inverting ill conditioned matrices. The solution
ofthe linear equations is iterated until the nonlinear governing equations are satisfied.
3.2.2.2 Flow Tangency and Influence Coefficients
The tangential flow equations are used to develop an influence coefficient matrix
to numerically solve flow tangency on the surface of the delta wing. The influence
coefficient matrix: is a matrix containing the linear governing equations that either fully or
partially define the solution depending on whether nonlinear governing equations are used.
If nonlinear equations define the system, the solution is only partially defined by using the
influence coefficient matrix. To complete the solution, the linear set of equation must be
iterated until the nonlinear governing equations are satisfied. This basic scheme can be
used for steady and unsteady solutions. The influence coefficient matrix: is the set of
equations that state that the velocity due to the sum of the singularity distributions on all
of the panels vectorially dotted with the normal panel vector must equal zero. In
mathematical notation, the boundary condition is satisfied at n points and is written as
also
n
Lqjj·nj=O
j=l
i = integers from 1 to n
where qij is jhe velocity vector induced by singularity distribution j on panel i. The
influence coefficients Rij and bij are the normal velocities induced on panel i by a vortex and
source distribution each of unity strength at panel j. The influence coefficients are
dependent entirely upon geometry of the wing. By calculating an influence coefficient
matrix using unity strengths, the unique solution of y and cr can be determined such that
50
the zero normal flow boundary condition is satisfied at all the collocation points.
However, this linear solution will not be unique if only the boundary condition influence
coefficients are specified.
To ensure that a unique linear solution exists, additional physical parameters must
be written that specifiy the circulation. The Kelvin condition ensures that the vorticity
generated on the wing is shed into the wake. Additionally, a Kutta condition for each
separation point must be included in the solution. For an inviscid model, a Kutta condition
enforces the global characteristics of separation.
3.2.2.3 Kutta Condition
By observation of the flowfield, it is assumed that the only effect of viscosity in the
problem is to fix the separation points at the leading edges. Each Kutta condition models
the effect of viscosity and forces a stagnation point to be located at the sharp point of each
leading edge. A Kutta condition can either by unsteady or steady. However, an unsteady
Kutta condition is nonlinear making its inclusion in the influence coefficient matrix
impossible if not linearized. In this effort, a steady Kutta condition is used because the
reduced frequencies characteristic in delta wing motions are low. The use of the steady
Kutta condition has been demonstrated previously by Katz and Plotkin57
. By the Kutta
condition being steady, the solution scheme is simplified because the steady equation is
linear whereas the unsteady Kuttacondition is nonlinear. The linearity of the Kutta
conditions allow them to be included in the matrix solution along with the equations
stating flow tangency. The difficulty in using an unsteady Kutta condition can be seen.
51
The circulation or vorticity is determined by a unique solution of the linear equations. If
this set of equations is not unique, then how can the proper circulation be found?
The stagnation point on each leading edge is modelled mathematicalJy by
observing that the vorticity at the leading edge must be zero. A corollary is the flow at the
leading edge must be irrotational; therefore, the tangential velocities on either side of the
leading edge must be equal. A second corollary states that the pressure must be equal on
either side ofthe separating shear in order for it to not sustain a load. Again the tangential
velocities must be equal for the pressures to be equal. The Kutta equations are as follows:
U =U
Pn/2 P(n/2+1)
Note that these velocities are local panel velocities. They must contain velocities due to
all components on the model, n source panels, n vortex panels of constant strength, 2
pot,ential vortices, and the free stream.
3.2.2.4 Kelvin Condition
The final equation that is included in the linear portion of the solution is the Kelvin
condition. The Kelvin condition ensures that the vorticity generated on the wing is shed
into the wake at each time step. The difficulty that arises with the Kutta condition is not
present since the Kelvin condition is linear for both the steady and unsteady cases. The
condition is represented by
52
n
"L!i·dli=rl+rr
i=1
The solution to the linear portion of the model is formulated in the following
manner. Flow tangency equations are placed in the first m rows of the matrix:. One
equation is written for each panel's collocation point The coefficients an through 3.n,m+3
are influence coefficients which when multiplied by the corresponding source or vortex
strength reflect the velocity due to nth source or vortex strength on the mth collocation
point. A linearized form of the kutta condition for the left and right leading edges to
ensure no shear force at the separation point. The Kelvin condition completely determines
the linear solution by specifying that the generated circulation is shed into the wake. The
right hand side (RRS) of the solution for the first n rows is simply the unsteady boundary
condition enforced at the collocation point. For the unsteady condition, the normal
velocity at each of the collocation points is specified to prevent flow through the
corresponding panel. The right hand side for the kutta conditions is remainder of the
terms from the linearized equation, namely the free stream velocities. The right hand side
for the Kelvin condition is simply zero.
Kelvin. Condition J l r/ J LRHS J n+ 3
rD n D," D,,"., lf~' 11 RHS, l
1l
x
l:U I
I D", Righl.Kutt:"Condilion D","., III (J"ryrn I I[ : 'I
Left. Kutta. Condition
 .........
53
3.2.2.5 Zero Force
An actual delta wing has a flow field such that a shear layer emanates from the
leading edge and follows a streamline. The shear layer must follow a streamline in the
flow since a shear layer cannot sustain a force. The shear layer separating from the
leading edge rolls up to form the primary vortices. For simplicity, the feeding sheet or
shear layer that lies between the leading edge and the free vortex is approximated as linear.
A depiction ofthis can be seen in Figure 3.4.
~
Fvl \....
\ Fsl
rr
~
.•._' Fvr
PSI I
s
Vinf" Sina.
+s y
Figure 3.4 Crossflow plane with branch cut forces
This model by itself violates the zero force condition as discussed previously, but when
coupled with the free vortex, the zero force condition can be enforced for both the feeding
54
sheet and the free vortex. A mathematical branch cut joins each leading edge of the wing
with the primary free vortex on the same side to make the solution of the flowfield
potential single valued. Zero force is obtained by summing the forces acting on each
branch cut as follows.
 
Fvl+Fsl=O
The forces which act at the center ofthe primary vortices are the forces due to the relative
velocity across the vortex line. The force is given by
rv=rI
and Utel is the relative velocity across the vortex tube. Since the cross product of the
direction denoted by i is zero when crossed with itself, only the crossflow components of
velocity are important. The components that are perpendicular can be seen in Figure
These components are perpendicular due to the conical assumption. The component in
the z direction is given by
Vz =V.,cosa tanSz =v.,cosa(:)
Likewise the component in the y direction is given by
Vy= V., cos a tan E)y = V., cos a(~)
Transforming to conical variables yields,
55
Vz = V..z* coso. tan E
Vy = V.y· cosa tan E
Vx~V",
Extending these equations to the unsteady case, the vortex itself is free to move; therefore,
the vortex motion must be subtracted to get relative velocity,
* •
Vz =V..z· cos a tan E _ '..(Z_v__z_O~)
ilt
where (yo,Zo) identifies the leading edge of the wing on the same side as the vortex,
Therefore,
and
[
* * • F. =pr V cosatanE(y*k+z*}) Cyv yo )k (zv zo*)J]'
v v co 6t.6.t
The remaining force is the unsteady force due to the discontinuity of potential across the
feeding sheet. This is computed by integrating the pressure differential across the feeding
sheet.
The unsteady change in pressure is
S6
Substituting the change in the coefficient of pressure into the equation above yields,
or
F [JZV( arvy arv) dzJ" [JYV( arvV arv s= zO  ax cocosa at P J+ yO  Ox oocosa,at) pdYJk
F
[( arv Y arv)( * *)..., ( arv y arv)C· *)k] s = p  Ox 00 cos a  at Zv  Zo J +  ax 00 cos a  at Yv  Yo
Equating the feeding sheet force to the force on the vortex yields two orthogonal
equations for each side.
In the y* direction,
In the z* direction,
( arvVcos a  arv )Cy * y *)+r [V y* cos a tan e _ (yv*yo·)] =0 Ox 00 at v 0 v· <Xl 6t
3.2.2.6 Steady Convergence
Since the zero force condition is nonlinear, the condition is satisfied outside of the
linear solution. The linear solution is iterated until the zero force condition on both pairs
,
I.
ll,
.~.I ~
".,
:: >
of feeding sheets with the potential vortex is satisfied. Zero force is satisfied by
57
computing the velocity at the old vortex position. This velocity is multiplied by the time
step to move the vortex to its new position. Iteration occurs until both vortices have
converged on the actual position. The convergence criteria is computed by calculating the
norm ofthe differences between the old and new positions.
norm=~(vpold (left)  Vp"ew (left))2 +(vpold (right)  vp""'" (right))2
A total of m+5 equations and m+5 unknows completely specify the problem where
m is the total number of panels used. The equations are as follows: n equations stating
zero normal flow, left Kutta condition, right Kutta condition, Kelvin condition, zero force
specification on the left feeding sheet and vortex, and zero force specification on the right
feeding sheet and vortex. With the problem being completely specified, the system of
equations is solved by using the nonlinear equations as a convergence criteria for an
iterative solution. Within each iteration, the linear equations are solved by LV
decomposition. This static solution is iterated to converge upon the correct primary
vortex positions. The noforce condition is applied after each solution within the iteration
step to predict the new refined position ofthe vortices.
3.2.3 Extension ofModel to Unsteady Motion
Extension of the model to include unsteady motion reqUires changes to the
boundary conditions as well as the zero force condition and pressure calculations.
I,
II,
:.1, ~
".,
". ~111
.~
58
3.2.3.1 Rigid Body Equation in Roll
Once the static solution to the wing rock problem has been found, the unsteadydynamic
coupled solution can be started. From the flow solution for the delta wing, the
initial roll moment is calculated for the wing. From rigid body dynamics, a roll velocity
can be calculated by assuming a value of inertia for the wing in roll. The dynamic
equation is as follows
The new roll angle is calculated by using a forward difference technique such that
;/" =;/, + Laero (/J.t)2
'f'new 'f'old I
"
3.2.3.2 Unsteady Boundary Conditions
(Eq.5) I
II,
~! Ii
:":i:l.
It is important to note again that the boundary conditions are different for the dynamic
solution. The noflow boundary condition used on the wing now has another velocity to
be taken into account. The resultant velocity normal to each panel still must be zero, but
the resultant velocity includes a term for the velocity of the body. The new boundary
condition is then stated as follows:
59
where qbody is the velocity of the wing itself In the static case, qbody is zero; therefore, the
dynamic boundary condition reduces to the static boundary condition.
3.2.3.3 Unsteady Zero Force
A similar solution technique to that used to find the static flowfield is then used to
find the proper vortex positions. The difference between the two techniques is that the
position of the vortices is found by using a forward difference in time technique instead of
an iteration scheme as in the first technique.
At each time step of the dynamic solution, a new roll moment must be calculated.
The roB moment is a function of the surface pressures, but the surface pressures are not
calculated using the static form of Bernoulli's equation. The dynamic pressure differs
from the static pressure because the potential of the model is not constant through time.
The potential, <fJ, changes with time by an amount d<fJ/dt. The equation for unsteady
pressure is as follows:
3.2.4 Unsteady Vortex Flow Control
A simple wing rock suppression technique has also been designed into the modeL
The bevels on the delta wing have been implemented so that they can be rotated about
their lower comers. This allows the use of the leading edges to simulate vortex flaps. The
vortex flaps modii)r the characteristics of the separation and therefore the primary vortex
position and strength. Furthermore, the position and strength of the vortices are crucial to
60
specifying the roll moment. Once the roll moment is affected in some way, any arbitrary
control algorithm can then be investigated to control th.e dynamic motion of the wing.
z
.5\='=====s~~~~~· ~/,; y
Figure 3.1 Delta Wing Cross Sectional Geometry with Vortex Flaps.
CHAPTER 4
RESULTS
Once the model of the wing rock phenomenon was developed, the model was
validated statically and dynamically against efforts by Arena49 and others. Separated
vortex control methods were implemented only after the model predicted the limit cycle
oscillation qualitatively. These methods demonstrate the flexibility ofthis type of model to
evaluate proposed control methods.
4. 1 Static Model Validation
For a delta wing at an angle of attack of 30 degrees and a sweep angle of 80
degrees, static tests were run to validate the model with experimental data obtained by
Arenal . The delta wing had a thickness to span ratio of 4.25%. All of the primary
flowfield characteristics have been qualitatively captured for conditions where vortex
breakdown is not present near the wing.
Previous ddta wing studies have shown static roll moments to be a function of roll
angle. A comparison of the inviscid delta wing model to experimental data can be made
by studying three fundamental variables that combine to result in a static ron moment for
the wing. These three variables are the lateral and normal positions of the primary
vortices and primary vortex strengths. In the comparison of lateral position versus roll
angle shown in Figure 4.1, the prediction of the upper vortex is closer to experimenta~
61
I.
i!!~
:::[1
62
2
"  Ld\ V'"""' .....I Code
1.5 "    Ilit!/< V""", P"",I Code
A " • Ld\ V""", E:q>. Rd"}2
A Ilillht V....... E>;>. Rd. 52
.... "
A A
A"'A
_. 
0.5 A ... A ...
y/S 0 • •
0.5 • • •••• •
1 •
•
1.5  •
2
60 40 20 a 20 40 60
$ (degrees)
Figure 4.1 Static Lateral Primary Vortex
Variation
value than the lower vortex where the upper vortex is the right vortex when the wing is at
a negative roll angle.
The normal position of the primary vortices is shown in Figure 4.2. The figure
shows qualitative agreement with experimental values; however, with the entire vorticity
assumed to be located at the center of vorticity, the computational values are shifted by a
bias of about 12% of the semispan. For static cases, the vortex corresponding to the
wing which is down moves closer to the wing. The variation of strength is shown in
Figure 4.3. These variations in the lateral and normal position and the strength of the
vortices combine to cause a variation in roll moment with roll angle. No data could be
found to compare against; however, the combination of these factors can be seen in
Figure 4.4. The sectional ro}) moment has been computed by integrating the effects ofthe
primary vortices on the wing. The effects are dependent upon the position and strength of
the primary vortices. The panel modd captured the experimental sectional roll moment
63
 Loft ......... Pond Cod<
 •  RqfJlV_PIIld Cod<
0.8 • Ld\ Von... Eop. Rtf. ~2
.. Ri8I1V_&l>.]u01
/"/
" •
0..7 .. , /
"
.... zls " 0.6 •
'.. . ,
0.5 • ..... .. "
• • • .... .... • .. 0.4
0.3
40 20 0 20 40
4> (degrees)
Figure 4.2 Static Nonnal Vortex Position
Variation
wen within the range of roll angles, 25° < <I> < 25°. Beyond ron angles of 25°or 25° the
model over predicts the experimental data because of the limiting assumptions used in
developing the modeL
The assumption that has the greatest effect is the inviscid flowfield assumption. By
adopting the inviscid assumption, the inertia forces are assumed only to be much greater
than the viscous forces. In other words, the Reynolds number is assumed to be high. As
the Reynolds number increases, the flow field transitions from laminar to turbulent. The
comparison to Arena's data shows the sensitivity of delta wing flows to Reynolds number.
Arena's data was obtained at a Reynolds number of 400,000. Previously, Hummel's52
flow regime at a Reynolds number of 900,000 was determined to be laminar. Thus, the
experimental flow is laminar for this case. Tbe result is that the flow does not have the
energy needed to traverse the adverse pressure gradient on the top surface of the wing.
Secondary s,eparation forms beneath the primary vortex on each side increasing the
,
Ih
'k
)
64
8
6
,,.............. 4
d
.'5' 2
III
*:s 0  Left Vortex >  .  .  ~ght Vortex
* 2 III
.'.......'
~
4 "
6 ,.
/
8
60 40 20 0 20 40 60
<P (degrees)
Figure 4.3 Static Primary Variation of Vortex
Strength
surf~ce pressure on the top of the wing. If the boundary layer was turbulent, the adverse
pressure gradient could be overcome keeping the pressure lower on the top surface of the
wing. So, the over prediction is due to the increased region of secondary separation since
the surface boundary layer is laminar. The model prediction would be closer to the case
where the boundary layer is turbulent. This is shown by the computational coefficient of
pressure distribution for a typical delta wing for roll angles of zero and fifteen degrees in
Figure 4.5. The computational pressure distribution agrees quantitatively with the
experimental data for the phi = 0.0 case. The experimental data was taken at an chord
station ofxJcr = 0.3. Notice the high suction peaks due to the primary vortices on the top
surfac,e of the wing. The higher suction peaks of the computational data are diminished in
the experimental data. The discrepancy can be attributed to the formation of secondary
vortices in the laminar experimental flow regime. As roll angle increases, the suction peak
on the side of the wing that is rotating downward has a higher coefficient of pressure due
6S
0.010
0.005
.,s..,  0.000 '"
r3
..10.005
0.010
!. ~'60oh Ii
••• •••
••• •• ••
..iQ.015
60 40 20 0 20 40 60
tj> (degrees)
Figure 4.4 Static Sectional Roll Coefficient Variation
to the change in position and strength of the vortex. The suction peak. on the opposite
side actually decreases denoting the corresponding vortex has moved further from the
surface of the wing as was shown in Figure 4.3.
The agreement of the various parameters fundamental to static delta wing flow
fields with experimental data demonstrated the feasibility ofthe model to capture the static
characteristics of the fundamental parameters. By combining these fundamental
parameters of position and strength of the primary vortices to form a static sectional roll
moment, the capability of the model is further corroborated by the agreement with
experimental static sectional roll moment in Figure 4.4.
~,
III
I~
66
2.5
  PhiaO.dqp<a
 Phi".0 dqp<a
• ~.!'IIlaO.O 2
1.5 """. .'" I ., / \ . ./
0.. 1
\
u \. / '" ./
0.5 ~ /
0
..:
0.5
1 0.5 0 0.5
y/s
Figure 4.5 Conical Pressure Distribution
4.2 Dynamic Model Validation
Once the model was vali.dated statically> the model was expanded to explore and
validate the dynamic characteristics. To ensure valid initial conditions, a steady state
solution was first found for a specific angle of attack and ron angle. Just as in the steady
case, the primary vortex position was iterated upon until no force existed on the linear
vortex sheets. When the correct static position was found, the wing was released in the
lateral mode to roll independently. The initial angular rate of motion at release was zero.
The other five degrees of freedom are fixed for any given case. The unsteady solution of
roll angle and vortex position and strength were calculated by integrating the unsteady
pressure over the wing and then marching the solution to the next solution in time by a
factor of.1.t, typically on the order of 0.05 seconds.
67
It is important to note that vortex breakdown on the wing is not necessary for the
wing rock limit cycle oscillation to occur. Arena and Nelson1,2.9 have shown that at higher
angles of attack where breakdown is present, additional damping is generated due to the
time lag in breakdown position. Currently, the model does not account for vortex
breakdown. Consequently, the dynamic model cannot accurately predict wing rock
amplitudes when any amount of vortex breakdown is occurring on the wing itself This
limits the model to angles ofattack less than 36 degrees for an 80° swept delta wing.
Upon release from a perturbed condition, the model demonstrated wing rock
lateral oscillations. The steady state amplitude of the wing rock osciUation is compared to
the experimental wing rock in Figure 4.6. When the computational envelope is compared
to experimental studies by Arena and Nelsonso, Nguyen, Yip, and Chambers53
, and Levin
and Katz54 the onset of the wing rock oscillations have decreased by an angle of attack of
80
9
70 Lombda 80 "'w
<> _ ... SUldy
A """"'."I1"d_Rd.H
;. 60 ..... NJlUY"' ...... Rd62
V'J Cl> .  Levin rd IC>U. Rd. 63 tb  (>  RodcJim, Rd: '3
ClJ 50
"0
''
Cl> 40 Ff "0 ....' ', .€
Q.. 30 ~ ~., I 't'
~ I J ''ilL. e 20 •I I .... '
I
... ,
10
~.....
I
o~!.J.J4l~w.ili.l.l..l.!..LlUJl.l..Lu.l.w~J.u..Il.w.u..J
10 15 20 25 30 35 40 45 50 55 60 65
ct (degrees)
Figure 4.6 Wing Rock Amplitude Envelope
68
10 degrees. Onset is described as the lowest angle of attack for which steady state limit
cycle oscillations occur. This bias in the onset may be due to the assumption that the
primary vortices are potential point vortices located at the center of vorticity. In other
words, actual experimental onset of oscillation may be delayed by the primary vortices not
being as coherent and strong as those modeled.
The envelope also shows a rapid decrease in the experimental envelope of Arena
and Nelson at an angle of attack of 36 degrees due to vortex breakdown. The present
computational effort as well as the "rocksim" effort made by Arena49 will not predict the
decrease in the amplitude because vortex breakdown is not modeled. This is regardless of
the bias in the computational efforts.
The case of a delta wing at 15 degrees angle of attack is demonstrated for sake of
qualitative comparison with experimental data. The experimental data is taken at an angle
of attack of 30 degrees~ however, the steady state amplitude of oscillation is close.
50
40 _
30
20
~ 10
ti 0
~
:;: 10
20
30
40
50
o 5 10 15 20 25 30 35 40 45 50
t (seconds)
Figure 4.7 Experimental Wing Rock Time
History
250 500 750 1000 1250 1500
t (seconds)
69
Investigation has shown that the model predicts a true limit cycle oscillation. The
converged wing rock amplitude will be the same regardless of the initial roll angle. Figure
4.7 shows the experimental wing rock time history for the roll angle amplitude. Given a
small disturbance at t=O, the wing rock oscillation builds to steady state. The steady state
amphtude ofthe motion is about 41 degrees in the experimental case.
Figure 4.8 shows the computational wing rock time history for the rock angle
amplitude. The inviscid model predicts that the steady state limit cycle amplitude to be
about 46 degrees.
The fundamental parameters of lateral and normal vortex position and primary
vortex strength were compared during a cycle of steady state oscillation. Figure 4.9 shows
the comparison of the experimental and computational dynamic lateral vortex locations.
The variations reveal a slight hysteresis in the lateral position over the majority of the
range of the roll angle traversed. Experimental data from Arena for an 80 degree deha
50
40
30
20
10
o
10
20
30
40
50 "''' ............
o
Figure 4.8 Computational Wing Rock Time
History
70
• Exp. Left Vortex, alpha;30, Ref.
52
• Exp. Right Vortex, alpha;30. Ref.
52
   Comp. Left Vortex, alpha; 15
40 20 0 20 40 60
Roll Angle, phi (degrees)
~ I
1==
!
4
,
~ '=
4
 J
.........
0
1
o
1.5
0.5
1.5
60
0.5
y/s
Figure 4.9 Dynamic Lateral Vortex Position
wing is also shown in Figure 4.9 The computational data is about the same order of
magnitude as the experimental data. The curves are similar to the data obtained from
Arena and Nelson's conformal mapping mode149 at an angle of attack at 20 degrees. A
direct quantitative comparison with the conformal data cannot be made due to the present
model having numerical instabiliti'es at extreme roll angles. The predicted steady state
amplitude of oscillation is about 80° as was shown in the plot of wing rock envelope. The
large amplitudes lead to numerical instability due to the interference of singularity
solutions between the wing panels and the primary potential vortices at 900 and 900 of
ron angle.
The normal variation in position of vortices for a cycle of steady state oscillation
exhibits hysteresis also. Normal variation is shown in Figure 4.11 The hysteresis for the
15 degree case is not as large as it is for the conformal model at an angle of attack of 20
degrees.
71
4  ~ , toCO.. ~
....
... r ""> ~ I
,(.,.''.'' ... ' ......::~
0.9
0.8
0.7
0.6
0.5
Z/S
0.4
0.3
0.2
0.1
o
60 40 20 0 20 40
Roll Angle, phi (degrees)
60
... EJiP. Left Primary Vortex:,
alpha=30, Ref. 52
• Exp. Right Primary Vortex:,
alpha=30, Ref. 52
   Comp. Left Vortex:, alpha=15
Comp. Right Vortex:, alpba=15
Figure 4.11 Dynamic Normal Vortex Position
The largest amount of hysteresis was found in the plot of vortex strength for a
steady state cycle. The hysteresis was about 12% of the semi span at _200 and 200 of roll
for the left and right vortices respectively. As the wing traversed positive roll angles, the
hysteresis occurs in the right vortex. At the same time hysteresis diminishes in the left
40 20 0 20 4Q
Roll Angle, ph~ (degrees)
........ : ,
,/ .....  ~~
f.. ""'"
1    Comp. Left Vortex ~
!Comp. Right Vortex
""~
~ ;:;'/
t'
4.0
6.0
fJ)
[Is 0.0
6.0
2.0
4.0
2.0
Figure 4.10 Dynamic Variation of Vortex Strength
72
vortex. The vice versa condition exists also. As the wing traverses negative roll angles,
the hysteresis is seen in the left vortex. The hysteresis in the right vortex decreases. No
experimental data could be obtained for comparison of primary vortex strengths; however,
the general shape of the curves matches that in the conformal mapping study although the
order of magnitude between the two studies is different. This difference cannot be easily
explained away by differences in angle of attack.
These three characteristics aU combine to provide a roll moment hysteresis. A plot
of roll moment versus bank angle is shown in Figure 4.13. The dominance of the
hysteresis in the normal position and strength of the primary vortices can be seen in
counter clockwise traversed lobes at the extent of the roll travel. The roll moment for the
experimental case characteristically has three loops to the hysteresis. As bank angle is
decreasing, the middle loop is traversed in a counterclockwise direction. This loop
essentially is adding energy to the system. As roll passes through an angle of 25 degrees,
the direction ofthe hysteresis loop reverses. This reversal provides damping, and the net
contribution of the outer lobes is a positive roll moment at negative roll angles and a
negative roll moment at positive roll angles. The reason why wing rock is limit cycle
oscillation at all due to the fact that the outer damping lobes naturally balance the inner
loop. The damping counteracts the energy added in the inner loop. If the outer lobes are
too small, the oscillation will diverge. If the outer lobes are just slightly larger than the
inner loop, the system is stable. So, two characteristics of roll moment must be present to
have the wing rock limit cycle oscillation. Hysteresis must be present in the roll moment
and the total areas ofthe hysteresis must be ,equal.
73
The computational hysteresis in Figure 4.13 shows the same characteristics as the
experimental hysteresis. The computational hysteresis has balanced loops just as the
0.02
0.01
C1 0
0.01
0.02
0.03 L...IJ....L...l...J...J...J....L..J...J....L..J.....Iu.....L.J..l....l...........L...IJ....L...l...J...J....I...w
60 40 20 0 20 40 60
4> (degrees)
Figure 4.12 Computational Roll Moment
Hysteresis versus Roll Angle
for a Steady State Cycle of
Wing Rock
0.1
0.08
0.06
0.04
0.02
q 0
0.02
0.04
0.06
0.08
0.1
50 40 30 20 10 0 10 20 30 40 50
~ (degrees)
Figure 4.13 Experimental Roll Moment
Hysteresis versus Roll Angle
for a Steady State Cycle of
Wing Rock
74
experimental case. The difference ~ies in the size of the inner loop; it is slightly larger. By
being slightly larger, the inner loop of the inviscid model is adding more energy to the
system. Therefore, the damping lobes must be larger to maintain steady state. As the
plots show, the damping lobes are elongated to compensate for the increase in energy.
With the model capturing the wing rock phenomenon qualitatively, the model can be used
within its own limitations confidently. Conse,quently, techniques now can be explored in
order to suppress the wing rock oscillation.
4.3 Static Vortex Flap Actuation
Since by observation the characteristics of the flowfield are dependent upon the
boundary condition at the separation points, a control methodology was developed based
on a vortex flap. A vortex flap modifies the boundary condition of the separation point by
actually moving the leading edge. Since the model was developed using a conical
assumption, flap must also be conical along the entire leading edge. The implementation
using the panel method is show in Figure 4.14, The bevel of the delta wing is rotated
about its lower corner while the connecting panel stretches to close the shape of the delta
wing body, This is meant to be representative of a leading edge flap deflected. The
<::: ....... .......  .......   \
\
\\
Figure 4, 14 Leading Edge Vortex Flap Geometry
75
vortex flap on each leading edge is constrained to rotate downward only. Due to
interference of the singularity panels, a maximum flap angle of 70 degrees was not
exceeded.
O.OlD
~
0.005 ..... , ,
\
q \
0.000
0.005 '
0.010
1/)=0 /) =0 I  ti =0: ti =60
\ ,
'.
30 40 50
0.a15 ''..........................................1........,.. '.......1....,,L....lJJ........L..........,
50 40 30 ·20 lD 0 10 20
ljl (degrees)
Figure 4.15 Static Roll Moment Versus Roll
Angle for Two Flap Angle
Combinations
To determine the exact effect of the leading edge vortex flaps, several test cases
were run while varying each vortex flap independently. A case showing the static change
in roll moment as the right flap is actuated is seen in Figure 4.15. As the right flap is
deflected and the left flap is held at zero flap deflection, the sectional roll moment curve
shifts down and to the left. Similarly, when the left flap is deflected while the right flap is
held constant, the roll moment curve shifts up and to the right. By actuating the flap on a
particular side, the roIl moment decreases in the direction ofthe flap. Given a wing that is
free to roll, the actuated flap makes the wing roll in the opposite direction of the lowered
flap.

76
The change in roll moment caused by a actuated flap is due to a combination of the
locations and strengths of the primary vortices changing. The variation of the lateralJ
position of the primary vortices with roll angle and flap defection is shown in Figure 4.16.
The flap deflection for each side affects the lateral location the most when that side of the
40 60
" "
Right

Left
 Ld\ Vortrx, J=O, ~c:o
 Rlpv....... 10. ,0
  Ld\Varttx. 1:::0. r a 60
  Ri&f';Vorta. leO.• r60
40 20 0 20
~. (degrees)
0.5
0.5
~
1
1,5
2 ..........
60
y/s
1.5
Figure 4.16 Lateral Vortex Position Variation
with Flap deflection
wing is down. For example, for a right flap deflection, the greatest affect is seen at
positive roll angles. In looking at the normal position versus roll angle plot of Figure 4.17
Nonnal Vortex Position Variation, the primary vortices are shifted down as the right flap
is deflected for the corresponding side. These characteristics of the flap deflections
indicate that modifying the angle at which the flow leaves the leading edges has potential
in controlling leading edge vortex flows and specifically in eliminating wing rock.
77
Left
';~
/" ...., . I
/ I \
/
 LeftVortex, 1=0. rO
 Right Vortex, 1=0. r=0
 LeftVon,,,, t=l). r=60
  Right Vortex. 1=0. r· 60
1
0.9
0.&
0.7
0.6
zJs
0.5
0.4
0.3
0.2
0.1
60 40 20 0 20 40 60
~ {degrees)
Figure 4.17 Nannal Vortex Positlon Variation
with flap deflections and roll angle
Comparison with steady state vortex slap investigation made by Rao m 1978
shows that for sev,eral vortex flap configurations, the current study performs within the
limited range of the model. The data agrees well both qualitatively as wen as
quantitatively. The normal force coefficient versus an angle of attack range from 0 to 32
degrees increases almost linearly from 0 to 1.7.
In Figure 4. 18, when compared to a number of flap configurations studied by
Rao19 which were deflected by 30 degrees, this study's normal force accurately predicts
the normal force generated on delta wing. The data splits the variance of the various flap
designs presented by Rao while the variance increases with angle of attack. Wing rock
data lies 18% below Rao's design with the highest data and 18% above the design with
the lowest normal force at an angle of attack of 24 degrees.
These comparisons show that the magnitude ofthe normal force coefficient is
correct for the 30 degree flap deflection case. The combination of primary vortex strength
78
and position is reasonable ev,en under the conditions ofthe limiting assumptions. Major
assumptions are
• Inviscid flow
• The center ofvorticity assumed to be concentrated at single point.
• The vorticity in the flow field was modeled by a point vortex
• The feeding sheet is assumed linear
• No vortex breakdown on the wing.
As the flap deflection increases the current study overpredicts the normal force
coefficient data as show in Figure 4.19. The overprediction could be due to a number of
factors. The primary vortex strength per degree of flap deflection may be incorrect for
larger flap deflections. The location of the primary vortices themselves may be incorrect
o 10 20 30 40
Angle of Attack, a (degrees)
+LEVF I, dhl=U7
_ LEVF I, dhlm2.S4
~LEVFI,dhlm3.81
~LEVF IV, dhlo.O
~LEVFI1l.dhlml.27
~LEVF II. <!hI"l.27 It;;ae'+i
~Rockllap
2
1.8
8 1.6 t 1.4
8'0 1.2
~ 1 ue 0.8
o~
0.6
l'l § 0.4
o
Z 0.2
o
0.2
10
Figure 4.18 Nonnal Force with 0=30.0
degrees
2
1.8
c:
u 1.6 ~ 1.4
.:;:;
............ [.2
~ 1 u
~ 0.8
.....
~ 0.6
E0.4
t 0.2
:z: o
0.2
 [0
~LEVFI. dhl9l.0
LEVF I, dhl1.27
~LEVF I. dhl=<2..l4
~LEVF I. dhl=1.27
~L1::VF I. dhl=<2..l4
+Ll::VF n, dbI=2..l4
tLEVF n, dbI=1.27
rockOa
o ]0 20 30
Angle of Attack, a (degrees)
40
79
Figure 4.19 Nonna1 Force with 0=45.0 degrees
due to amalgamating all the vorticity at the center of vorticity and modeling it with a point
vortex. Furthermore, the feeding sheet panel on which a "zeroforce" condition is
enforced is assumed linear may not be truly linear.
4.4 Dynamic Vortex Flap Actuation and Control
Using an active control scheme, the vortex flaps were implemented with a
derivative feedback loop into the single degree of freedom model. When the wing has a
positive ron rate, the right flap is deflected by a proportion ofthe roll rate. When the wing
has a negative roll rate the left flap is deflected by a proportion of the roll rate. The
control laws are as follows:
o = K· ¢ if ¢ > 0
r
0, = K .¢ if ¢ < 0
80
where K is the gain on the feedback [oop. For this case, a = 15°, A = 80°, the gain is set
to a value of 1.5 (K = 1.5) The flaps are limited to a maximum deflection of 70° to
prevent interference of the singularity solutions.
Promising effects can be seen immediately by implementing this feedback loop.
Feedback control decreases the amplitude by about half within two cycles of the control
laws being turned on. In Figure 4.20, the amplitude decreases from 47 degrees to about
20 degrees within two cycles of the control being turned on. The time to half amplitude is
about 50 time steps.
60
50 r
40 r
30 
20
10 .p 0 ~\jVv
10
20
30
40 Control
50 On
60 I h I I I I
0 50 100 150 200 250 300 350 400
t*
Figure 4.20 Wing Rock Suppression
81
1LeftAap I  RiBhtFlap
100 200 300 400 500
t*
16
14
12
10 
8 8
6
4
2
0
0
Figure 4.21 Flap Deflection History During
Control
The flap deflection history is shown for the same control cycle in Figure 4.21. This
control method is nonlinear since the flaps are not allowed to rotate up. The maximum
deflection of the flaps occurs when the control is first turned on. The flap deflections
quickly diminish revealing that the roll rate is indeed being affected. The total
0.02 .,  .,  .. ~ .,  .
0.01
CI 0
"
_ __ ..
~ : .!\fVvvv~
0.01
0.02 r' .
, , , ,
_ •• __ ••, ••• __ •••• __ •• 1, _ •• __ •••• __ •• ,, ,,
,, ,, , ,
, ,
, , . _., _., _._.__ _ ..
,, ,, ,, , ,,
, , ,
200 300 400
t*
100
0.03 iii .....lI
o
Figure 4.22 Ron Moment Time History During
Control
82
computational time needed to investigate a control methodology, is about 5 hours on a
time shared mM RS6000 workstation. This time includes letting wing rock build to
steady state about a total of 80 cycles and then damping the oscillation by 90% in about 8
cycles. A total of 50,000 time steps were needed to resolve the build up and damping of
the wing rock oscillation.
Observation of the parameters that combine to drive the wing rock oscillation
demonstrates the ability to not only control the combination of the parameters, but the
individual parameters as well. Each parameter, y/s, zis, rls, is controlled and converges to
a finite value. As steady state damped conditions are reached, the fluctuations of the
lateral and normal positions and vortex strengths are not merely canceling each other, but
the deviations ofthe values are diminishing. Significant damping is accomplished within 2
cycles ofcontrol being turned on.
In Figure 4.23 the characteristic steady state oscillation is shown along with the
spiraling lateral vortex position after control was turned on. The lateral vortex position on
1.1 r, 0.75
1.05
0.95
Yrighls
0.9
0.85
0.8
0,8
0.85
0,9
Ylerr's
0.95
1
1.05
1.1
20 0 20 40 60
«I> (degrees)
0.75 ..........................................L..L...JL..L...JL....I....JL....I....JL....l...JL....I....JL....l...J'''
60 40
Figure 4.23 Dynamic Lateral Vortex Position during Control
83
0.5 ..,
0.45
0.4
tis 0.35
0.3
0.25  _. Left Vortes
 Right Vortex
20 0 20 40 60
<p (degrees)
0.2 L..L...L...L...L...L...''''''L...J...JL...J...JL...J...JI...LJI...LJ..........
60 40
Figure 4.24 Dynamic Normal Vortex Position
during Control
each side oscillates about the point <1>=0., Iy/sl = 0.825 until finally converging after about
six cycles.
Figure 4.24 shows the left vortex departing from the steady state loop better. The
normal position spirals counterclockwise until converging at zls=0.375. The right vortex
is similar except that the spiral motion occurs in the clockwise direction.
The control algorithm affects the vortex strength immediately by increasing the
strength in the respective directions. The flaps gradually diminish the strength until
converging upon a value of ly/sl=5.4.
84
The exact effect ofthe flap is best seen in Figure 4.26. During the last cycle of the
wing rock oscillation before control is turned on, the cycle still exhibits the three major
loops as seen previously. When control is turned on, the roll moment decreases in a
counter clockwise spiral toward zero moment. The counter clockwise spiral denotes a
stable dynamic system. The roll moment decreases as the roll angle decreases until the
motion is damped out completely. Thus, a promising technique to alleviate wing rock is
one that is able to dynamically move the points of separation with respect to the main
wmg.
6.0 1.0
!
4.0 i 3.0
YIett's 1 Yrigbtf'
2.0 5.0
7.0
20 0 20 40 60
cjl (degrees)
0.0 L.J....l.I...l.I...l.I...l.I...L......L...JL......L...JL...J....L...J....L.L....JL...J....L...l...J
60 40
Figure 4.25 Dynamic Vortex Strength during Control
85
0.02
0.01
o 0.00
0.01
0.02
0.03 ........u.o.J.ju.u.r..u..u..LJ...J..l.u..u............u.u.r............LJ...J..l...................
50 40 30 20 10 0 10 20 30 40 50
~ (degrees)
Figure 4.26 Roll Moment versus Roll Angle
Stability Curve
4.5 Projected Control Methodologies
A panel method was used so tha~ arbitrary geometries and control techniques
could be implemented to investigate the alleviation of wing rock. Spanwise blowing and
vortex control flaps are two methods proposed by previous investigators to control the
oscillation. The hyst,eresis of the vortices.' location and strength has been controlled by a
leading edge flap in this investigation by moving the separation point with respect to the
wing enough to control the location of the vortices. Arena49 documents that a distinctive
characteristic of vortex behavior on delta wing during wing rock is that hysteresis in
position is confined to the direction normal to the surface of the wing. This suggests that
wing rock will be able to be controlled through blowing on the upper surface of the wing
since blowing is able to move the separation point of the emanating shear layer. This is in
fact the conclusion at which Wong36 arrives by implementing a dynamic feedback system
86
alone. Wong is able to attenuate the wing rock motion significantly. This does not prove
that the source of wing rock is eliminated, however. By approaching the problem from a
coupling of the aerodynamics with the vehicle dynamics, a better knowledge of the physics
is attained, and therefore, the physics can be exploited to its fullest extent.
CHAPTER 5
CONCLUSIONS AND RECOMMENDATIONS
5. I Conclusions
In the investigation of separated vortical flows, this effort has demonstrated the
flexibility of an inviscid panel method to capture the primary characteristics of a dynamic
separated vortical flow field quickly. Specifically, this study addressed a solution to a
nonlinear flow field I vehicle dynamic instability known as wing rock. Several aspects of
the model make it useful in primary parameter identification as wen as in an iterative
search for control solutions to separated vortical flows. The aspects which are important
are the simplifYing assumptions, ease of modeling a given geometry and flow field, and the
execution time required to predict a dynamic solution.
The simplifying assumptions of the problem definition were balanced against the
ease of geometry dis,cretization, flow field accuracy, and type of information desired.
Since primary characteristics of the flow field were desired, flow field accuracy was not as
important as the ease of modeling the geometry and the rapidity of the solution.
Simplifying assumptions such as an inviscid, conical flow field allowed the problem
geometry to be reduced to 2D and flow field to be reduced to a quasi3D scenario.
Additionally, the panel method used in this study allowed the 2D geometry to be
easily modeled. The leading edges of the delta wing were modeled with a 45 degree
87
88
comer so that separation could be accurately placed at the leading edges. Wing thickness
could also be modeled as long as the singularity solutions on each pand were not close to
each other. The gridding of the geometry was accomplished quickly so that the geometry
could be repaneled at every time step. During the dynamic solution, any geometry
motion such as the flaps was handled by repaneling at every time step.
The flow field solution was detennined by using singularity solutions of the
inviscid flow field on each panel of the geometry. Given the linear nature of the steady
state solution, the static singularity solutions were used in matrix form to determine the
static portion of the solution. The dynamic piece of the solution was then added to the
static solution to detennine the total solution. The dynamic piece of the total solution was
potentially a very time consuming, as the potential ofthe flow field changes with time and
must be integrated from the far field where the flow field is not changing.
Validation of