ABSTRACT
Title of dissertation: Unsteady force production on a flat
plate wing by large transverse gusts
and plunging maneuvers
Gino M. Perrotta, Doctor of Philosophy, 2017
Dissertation directed by: Professor Anya Jones
Department of Aerospace Engineering
The transient forces produced by large-amplitude transverse gust encounters
and plunge maneuvers were studied experimentally in a water-filled towing tank.
Forces were measured as a flat plate wing with an aspect ratio of four was towed
through a fluid gust and as the same wing performed plunge maneuvers which
matched the shape of the gust velocity profile. The transient velocity in each case
conformed to the sine-squared profile, and the peak transient velocities were of the
same order of magnitude as the steady towing velocities. In most cases, the wing
pitch angle was high enough to cause constant flow separation. Even at low wing
pitch angles, the increase in flow incidence angle by the transverse gust or plunge
velocity was enough to cause flow separation. Transient force magnitudes were shown
to increase with increasing stream-normal velocity for both the gust encounters
and plunge maneuvers. Transient forces varied with increasing wing pitch angle
during gust encounters but not during plunge maneuvers. Force histories in each
case were compared to predictions made by existing small-perturbation force models,
and adaptations were made to those models based on physical interpretation of the
observed characteristics. Measured forces in both the gust encounters and the plunge
maneuvers were found to correspond more closely to predictions made based on
attached flow than on separated flow, which supports the suggestion that the presence
of a leading edge vortex significantly augments the transient lift. Additionally, a large
trailing edge vortex forms at the end of the gust encounter which temporarily reduces
the force production below the steady-state values. This was not observed in the
plunge maneuver force histories, which were much closer to quasi-steady than were
the gust encounter force histories. This analysis contributes to the understanding of
unsteady force production in large-amplitude events, and in particular in conditions
with separated flow, the behaviors of which are not adequately captured by existing
small-perturbation models.
Unsteady force production on a flat plate wing by
large transverse gusts and plunging maneuvers
by
Gino M. Perrotta
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2017
Advisory Committee:
Professor Anya Jones, Chair/Advisor
Professor Ken Kiger, Dean’s Representative
Professor James Baeder
Professor Stuart Laurence
Professor Imraan Faruque
© Copyright by
Gino M. Perrotta
2017
Acknowledgments
I am fortunate to have been a part of the Separated and Transient Aerodynamics
Laboratory (STAL) under the mentorship of Professor Anya Jones since the lab’s
early days — before its clever “back-ronym” of a name, before its growth into
double-digit membership, before there were even walls around the work space. By a
combination of logistical necessities and Anya’s generous trust, I was given greater
responsibility than my (lack of) seniority warranted, which — along with Anya’s
patient guidance — fostered both my competence and confidence as a research
engineer. Now at the end of my years with STAL, I would like to recognize what a
special opportunity this was, and to thank Anya for inviting me to be a part of the
team.
Gratitude is also owed to my laboratory peers. In particular, thanks to Dr.
Andrew Lind, who with enthusiasm and diligence set such a high standard that all
of us at STAL were improved by its pursuit. Thanks to the soon-to-be-Drs. Peter
Mancini and Field Manar, who joined the lab at the same time that I did and so with
whom I shared the effort and accomplishment of each academic milestone. Their
assistance, insight, and friendship were central to my graduate school experience,
and are greatly appreciated.
These years have not been all work, and I could not have succeeded here
without the neighborly academic community I found myself a part of. Thanks to
the Lees, the Myerses, the Schneiders, and the Kordells for their encouragement and
inspiration.
ii
Thanks especially to Mom and Dad, who supported me in uncountable ways
both during and long before my academic years. Science fair projects and after-school
engineering programs were as much a part of my education as was any class, and I
am grateful to my parents for seeing the value in those early projects. Thanks also
to my siblings, Robert and Michael, to my grandparents, aunts, uncles, and cousins,
and more recently also to my in-laws for optimistic reassurance throughout my time
in school.
Most of all, thanks to my wife, Abby, whose constant love and stubborn belief
in my abilities has sustained my own perseverance. Thanks for always supporting
me, for being the test audience for every practice presentation, and for being my
greatest fan and best friend.
iii
Table of Contents
List of Tables vi
List of Figures vi
Nomenclature viii
1 Introduction 1
1.1 Steady aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 Thin-airfoil theory . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Joukowsky airfoils . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Empirical extensions . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.4 Quasi-steady analysis . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Unsteady solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 Impulse response . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Sharp-edged gust . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.3 Harmonically oscillating wings . . . . . . . . . . . . . . . . . . 9
1.2.4 Arbitrary wing motion . . . . . . . . . . . . . . . . . . . . . . 10
1.2.5 Indicial theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Empirical adaptation . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.1 Streamwise perturbations . . . . . . . . . . . . . . . . . . . . 17
1.4.2 Transverse perturbations . . . . . . . . . . . . . . . . . . . . . 18
1.5 Present work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Methods 22
2.1 Towing tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Test article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 Gust generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Gust encounter kinematics and parameter matrix . . . . . . . . . . . 28
2.5 Plunge maneuver kinematics and parameter matrix . . . . . . . . . . 30
2.6 Force measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6.1 Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6.2 Dynamic calibration . . . . . . . . . . . . . . . . . . . . . . . 39
iv
2.6.3 Lift normalization . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.7 Flow velocity measurements . . . . . . . . . . . . . . . . . . . . . . . 42
2.8 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Force production in transverse gust encounters 46
3.1 Gust encounter force history features . . . . . . . . . . . . . . . . . . 46
3.1.1 PIV flow velocity measurements . . . . . . . . . . . . . . . . . 49
3.1.2 Trends with respect to gust parameters . . . . . . . . . . . . . 53
3.2 Force history predictions . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.1 Adaptations of indicial theory . . . . . . . . . . . . . . . . . . 58
3.3 Variations in peak force production . . . . . . . . . . . . . . . . . . . 65
3.3.1 Unsteady analysis of peak force . . . . . . . . . . . . . . . . . 69
3.4 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4 Plunging wing force production 72
4.1 Plunging wing force history features . . . . . . . . . . . . . . . . . . . 72
4.2 Trends with respect to plunge kinematics . . . . . . . . . . . . . . . . 75
4.2.1 Effects of varying Reynolds number . . . . . . . . . . . . . . . 76
4.2.2 Effects of varying plunge height . . . . . . . . . . . . . . . . . 77
4.2.3 Effects of varying plunge width . . . . . . . . . . . . . . . . . 80
4.2.4 Effects of varying wing pitch angle . . . . . . . . . . . . . . . 81
4.3 Force history characterization . . . . . . . . . . . . . . . . . . . . . . 87
4.3.1 Two-term decomposition . . . . . . . . . . . . . . . . . . . . . 87
4.3.2 Comparison to analytical models . . . . . . . . . . . . . . . . 91
4.4 Comparison to gusts . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.4.1 Inertial forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.4.2 Velocity distribution . . . . . . . . . . . . . . . . . . . . . . . 106
4.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5 Summary and conclusions 110
5.1 Key gust encounter conclusions . . . . . . . . . . . . . . . . . . . . . 111
5.2 Key plunge maneuver conclusions . . . . . . . . . . . . . . . . . . . . 112
5.3 Highlighted original contributions . . . . . . . . . . . . . . . . . . . . 114
5.4 Suggestions for future work . . . . . . . . . . . . . . . . . . . . . . . 114
v
List of Tables
2.1 Gust encounter test matrix . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 Plunge maneuver test matrix . . . . . . . . . . . . . . . . . . . . . . 33
4.1 Plunge maneuver steady to unsteady force ratio . . . . . . . . . . . . 85
4.2 Plunge maneuver inertial to total unsteady force ratio . . . . . . . . . 85
List of Figures
1.1 Joukowsky airfoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Wagner’s and Ku¨ssner’s unsteady lift solutions . . . . . . . . . . . . . 8
2.1 University of Maryland towing tank . . . . . . . . . . . . . . . . . . . 23
2.2 Flat plate wing in CAD . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Flat plate wing photo . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Static force coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Gust generation system . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.6 Gust velocity profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.7 Gust encounter kinematics . . . . . . . . . . . . . . . . . . . . . . . . 29
2.8 Plunge velocity profile . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.9 Plunge maneuver kinematics . . . . . . . . . . . . . . . . . . . . . . . 32
2.10 Plunge motor velocities as prescribed and performed . . . . . . . . . 34
2.11 Force data post-processing . . . . . . . . . . . . . . . . . . . . . . . . 35
2.12 Data collection synchronization . . . . . . . . . . . . . . . . . . . . . 37
2.13 Force data dynamic calibration . . . . . . . . . . . . . . . . . . . . . 39
2.14 Alternative lift normalization example . . . . . . . . . . . . . . . . . 41
vi
2.15 PIV measurement setup . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.16 PIV field of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1 Gust encounter force histories . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Gust encounter flow velocities for α = 45° and G = 1.0 . . . . . . . . 51
3.3 Local vorticity during a gust encounter at α = 45° and G = 1.0 . . . . 52
3.4 Gust force production with respect to wing pitch angle . . . . . . . . 54
3.5 Gust force production with respect to gust velocity ratio . . . . . . . 55
3.6 Quasi-steady gust force predictions . . . . . . . . . . . . . . . . . . . 58
3.7 Models of gust encounter force production . . . . . . . . . . . . . . . 64
3.8 Gust force characterization by peak force . . . . . . . . . . . . . . . . 66
3.9 Comparison of peak gust encounter forces to quasi-steady predictions 67
3.10 Convergence of peak forces for gust encounters with different initial
conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.11 Comparison of peak gust encounter forces to unsteady predictions . . 70
4.1 Plunge maneuver force histories . . . . . . . . . . . . . . . . . . . . . 73
4.2 Plunge force production with respect to freestream Reynolds number 76
4.3 Variations in total dynamic pressure with respect to plunge velocity
ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4 Plunge force production with respect to maneuver height . . . . . . . 79
4.5 Plunge force production with respect to maneuver width . . . . . . . 81
4.6 Plunge force production with respect to wing pitch angle . . . . . . . 82
4.7 Additional cases of plunge force production with respect to wing pitch
angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.8 Best fit of the two-term lift decomposition for a high velocity ratio
plunge maneuver: w = 1c, h = 1c . . . . . . . . . . . . . . . . . . . . 90
4.9 Best fit of the two-term lift decomposition for a low velocity ratio
plunge maneuver: w = 6c, h = 3c . . . . . . . . . . . . . . . . . . . . 91
4.10 Characterization of plunge force production for all cases . . . . . . . . 92
4.11 Comparison of measured and predicted lift for velocity-proportional
C1 term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.12 Comparison of measured and predicted lift for acceleration-
proportional C2 term . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.13 Plunge force predictions for w = 1c, h = 1c, α = 0° . . . . . . . . . . 100
4.14 Plunge force predictions for w = 6c, h = 3c, α = 0° . . . . . . . . . . 101
4.15 Plunge force predictions for w = 1c, h = 3c, α = 20° . . . . . . . . . . 102
4.16 Comparison of plunge maneuver and gust encounter forces for α = 0° 103
4.17 Comparison of plunge maneuver and gust encounter forces for α = 20° 104
vii
Nomenclature
A aspect ratio, l
c
c wing chord
C(k) Theodorsen’s function
CD drag coefficient
CL lift coefficient
CN normal force coefficient
CT tangent force coefficient
C1 empirical velocity-proportional force coefficient
C2 empirical acceleration-proportional force coefficient
D drag
Dnorm drag normalized by mean kinetic energy
FN normal force
FN, norm normal force normalized by mean kinetic energy
FT tangent force
FT,norm tangent force normalized by mean kinetic energy
f frequency
G gust velocity ratio, Vg
U∞
g acceleration due to gravity
h plunge maneuver height
k reduced frequency, fc
2U∞
L lift
viii
Lnorm lift normalized by mean kinetic energy
l wing span
q∞ freestream dynamic pressure, 12ρU
2
∞S
Re Reynolds number, U∞c
ν
R least-squares residual
S wing planform area, cl
s semi-chord lengths, 2tU∞
c
T duration
t clock time
t∗ convective time, tU∞
c
U∞ freestream velocity
Umax peak total velocity,
√
U2∞ + V 2
Vg peak gust velocity
Vp peak plunge velocity
w plunge maneuver width
wg(x) gust velocity distribution
z complex coordinates, z = x+ iy
ix
Greek symbols
α wing pitch angle above horizontal
Γ circulation
γ vorticity distribution
ζ complex coordinates, ζ = χ+ iη
θ gust incidence angle, tan
(
wg
U∞
)−1
λ gust travel velocity fraction in Miles’s function
ν fluid kinematic viscosity
ρ fluid density
σ integration variable to s
φ Wagner’s function
ψ Kussner’s function
List of Abbreviations
AVT Applied Vehicle Technology
CAD Computer-Aided Design
CNC Computer Numerical Control
MAV Micro Air Vehicle
PID Proportional Integral Derivative
PIV Particle Imaging Velocimetry
UDP User Datagram Protocol
x
Chapter 1: Introduction
Unsteady aerodynamic force production is a topic of interest in many
applications. The proper control of an aircraft (and the comfort of its occupants)
flying through a gusty environment, in which the local flow velocities vary over time
and location, depends on the quantification of the forces produced by the varying
flow conditions [1]. The efficiency of rotors in helicopters and wind turbines are
significantly affected by the unsteady forces related to dynamic stall [2]. Micro air
vehicles (MAVs) are particularly vulnerable to atmospheric gusts due to the relative
scaling of steady and transient velocities. Additionally, flapping wing MAVs rely on
unsteady aerodynamics for lift, thrust, and control and so require unsteady force
quantification in their design [3]. It is these MAV applications in particular which
lead the current work to study force production of large-amplitude unsteady events.
The decreasing size of practically-realizable micro air vehicles has increased
interest in unsteady force models that do not assume small perturbations [3–6].
The primary difficulty of large-amplitude unsteady phenomena is that they cause
flow separation, which greatly increases the complexity of the flow around the wing
and leads to unsteady and non-linear forces. The current work investigates the
aerodynamic force production of large-perturbation, two-dimensional, unsteady flows;
1
it is therefore useful to review the particular branch of analytical aerodynamic models
that addresses various forms and degrees of flow unsteadiness.
1.1 Steady aerodynamics
The first quantifications of force created by lifting bodies were for various steady
flow conditions. The assumption that neither the body nor the flow characteristics
were changing in time allowed for the intentional neglect of several complicating
features, such as vorticity in the wake of the wing and the inertial reactive force
caused by accelerations.
1.1.1 Thin-airfoil theory
The simplest of lifting bodies is the idealized thin, flat, infinite span (for an
effectively two-dimensional problem) plate. If the flow is further assumed to be
incompressible and inviscid, then the complex effects of flow separation, often called
stall, can be removed. By analytically finding the vorticity distribution over the
wing that is required to prevent flow passing through the wing, the classical solution
of thin-airfoil lift coefficient can be found:
CL ≈ 2pi sin(α) (1.1)
Where α is the wing pitch angle above horizontal. This solution requires that the
flow is steady, inviscid, and incompressible, and that the wing is thin, flat, and very
nearly aligned with the flow (α ≈ 0). The first restriction that can be loosened is
2
the requirement that the wing be flat. The linearizing small-angle approximation
which is commonly applied to the wing chord line (through α) can also be applied
to the wing camber line to extend this solution to wings that are not exactly (but
still very nearly) flat. By again finding the vorticity distribution which enforces no
flow through the surface of the wing, it is found that minor camber of a thin airfoil
causes a shift in the zero-lift angle of attack but does not change the slope of the lift
coefficient with respect to that angle [7].
1.1.2 Joukowsky airfoils
The idealized thin wings of the previous section can be made more physically
representative by incorporating structural thickness. Conformal mapping is used to
transform the shapes of aerodynamic bodies into shapes more readily formulated
by potential flow elements. The Joukowsky (or Zhukovsky) transform is a common
mapping between a circle and an airfoil shape. The surface of the airfoil is represented
by a path in the coordinates z = x + iy, which is transformed to a circle in the
corresponding coordinates ζ = χ+ iη by the equation:
z = ζ +
1
ζ
(1.2)
A circle of radius a in the ζ-plane which is centered on the origin corresponds to a flat
plate of chord 4a in the z-plane. Moving the circle center off of the origin introduces
thickness and camber to the z-plane lifting body. In this way, the relatively simple
potential flow solutions for a circle can be used to investigate the aerodynamic
3
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
x
-0.5
0
0.5
1
1.5
2
2.5
3
y
Flat
Thick
Camber
-1 0 1
-1
0
1
Figure 1.1: Various Joukowsky airfoils and the corresponding ζ-plane circles.
qualities of physically-realizable airfoil shapes.
It is worth emphasizing that the Joukowsky transform can create many z-plane
shapes which are not airfoils; aerodynamic application of this equation requires
that the ζ-plane circle intersect the point ζ = 1. Several Joukowsky airfoils are
demonstrated in figure 1.1.
1.1.3 Empirical extensions
A common compromise in the quantification of aerodynamic force production
is the application of empirical data as an adaptation or replacement for an analytical
model. Thin-airfoil theory cannot address stalled flow over a wing, but its general form
of lift as a dimensionless coefficient multiplied by dynamic pressure can be extended
to a semi-empirical model by substituting lift coefficient values from observations.
4
This system of measurement and careful interpolation allows for the quantification
of forces during stall and for bodies of arbitrary shapes. Anderson [7] compares
the simplified analytical solution for drag on a cylinder (potential flow predicts no
pressure drag in steady flow — d’Alembert’s paradox) to the measured results for
a range of Reynolds numbers. The proper scaling of these empirical coefficients is
perhaps the best method of predicting drag on cylinders.
1.1.4 Quasi-steady analysis
One step towards an unsteady flow solution is quasi-steady analysis. In this
method, the flow and body conditions are allowed to change over time, but the
force production at any time is calculated using steady-flow formulas based on the
characteristics at that particular moment. Any changes in the flow are assumed to
immediately develop to the steady-state condition, so no time delay is present in the
force production and the events in the flow history are not accounted for.
Quasi-steady analysis introduces the ability to capture inertial force production;
the instantaneous flow characteristics can include an acceleration, which accounts
for forces not present in steady conditions. For a thin, flat plate [2, 8]:
F⊥ = −1
4
piρc2lV˙⊥ (1.3)
Where F⊥ is the force normal to the wing, V˙⊥ is the wing-normal component of the
acceleration of the wing, c is the wing chord, and l is the wing span. This inertial
force is not due to the mass of the wing — that force is separate and is removed from
5
aerodynamic force considerations — it is due to the acceleration of local fluid, which
increases the apparent inertia of the wing [9]. This is called the “apparent-mass”
or “added-mass” force. Only chord-normal acceleration causes this effect — any
chord-wise component of acceleration causes no added-mass force, and no added-mass
force is oriented chord-wise. For a wing which can accelerate horizontally or vertically
(surging and plunging):
Cl, inertial =
F⊥
1
2
ρU2∞cl
cos(α) (1.4)
V˙⊥ = U˙ sin(α)− V˙ cos(α) (1.5)
Equation 1.3 can be expanded and normalized by equations 1.4 and 1.5:
∴ Cl, inertial =
pic
2U2∞
(
U˙ sin(α)− V˙ cos(α)
)
cos(α) (1.6)
This expresses the inertial contribution to lift, Cl, inertial, as a linear scaling of the
chord-normal projections of the surge and plunge accelerations, U˙ sin(α)− V˙ cos(α).
Since the added-mass force is chord-normal, the lift component requires an additional
cos(α) projection.
1.2 Unsteady solutions
In practice, changes in the flow take time to develop into a new steady state.
By Kelvin’s Circulation Theorem, any change in circulation around the wing must
be matched by opposing circulation shed into the wake [7]. The presence of this
vorticity in the wake affects the forces on the wing. As the wake departs farther from
6
the wing, this effect is decreased and eventually becomes negligible. This gradual
change in force production in response to instantaneous changes in flow conditions is
the core concept for modeling of unsteady aerodynamic force production.
1.2.1 Impulse response
A foundational work on unsteady force production is the analytical solution
to small-perturbation impulsive (as in a Heaviside “step” function) lift changes by
Wagner [10]1. Wagner’s solution describes the temporary lift deficiency caused by
vorticity introduced in the wake of the wing by the change in circulation. The
influence of the wake vorticity decreases as the wake is left behind the translating
wing, and so the lift asymptotically approaches the steady-state condition. As in
steady thin-airfoil theory, Wagner’s solution requires the assumption that the wing
and wake are very nearly aligned with the flow. It also retains the assumptions of
inviscid, incompressible, and two-dimensional flow.
Wagner’s function is typically represented by φ(s), where s = 2tU∞
c
is a
normalized time, equivalent to twice the number of chord-lengths traveled by the
wing. The function cannot be written in a regular algebraic form, so it is either
tabulated or approximated by the following equation [8]:
φ(s) ≈ 1− 0.165e−0.0455s − 0.35e−0.3s (1.7)
Wagner’s function is plotted in figure 1.2.
1Wagner’s and Ku¨ssner’s original papers are in German; their results are described in English in
a later aeroelasticity textbook by Blisplinghoff et al. [8].
7
0 5 10 15 20 25 30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 1.2: Wagner’s and Ku¨ssner’s unsteady lift solutions.
Jones [11] extended Wagner’s work by adding tip vortex effects. Jones’s
solution varies with respect to wing aspect ratio, and suggests that the transient
force production is less affected by the tip vortices than is the steady-state lift.
1.2.2 Sharp-edged gust
Ku¨ssner [12] produced an analytical solution for the unsteady lift on a wing
entering a sharp-edged transverse gust. While Wagner’s “gust” reaches the whole
wing simultaneously, Ku¨ssner’s gust progresses from the leading edge to trailing edge
of the wing.
As with previous results, Ku¨ssner’s solution relied on the assumption that the
gust velocity was much smaller than the steady forward velocity of the wing so that
the effect of the gust was only a small perturbation to the steady conditions. It was
8
also assumed that the flow is always attached to the suction surface of the wing,
which is mainly dependent on the same small perturbation limit. Ku¨ssner’s function
is (as is Wagner’s) an equation that describes the temporary deficiency in lift caused
by the circulation in the wake of the wing. Ku¨ssner’s function is typically represented
by ψ(s), which can be approximated by [8]:
ψ(s) ≈ 1− 0.500e−0.130s − 0.500e−s (1.8)
Ku¨ssner’s function is plotted in figure 1.2.
As an extension of Ku¨ssner’s solution, Miles [13] solved for the forces caused
by a sharp-edged gust that is traveling through the fluid (in addition to the wing
moving through the fluid). Miles’s function falls somewhere between the solutions
of Wagner and Ku¨ssner (see figure 1.2), depending on a velocity factor describing
the rate of travel of the gust relative to the wing, λ−1 = 1− Ugust
Uwing
. When the gust
is stationary, λ = 1 and the solution converges to Ku¨ssner’s function. When the
gust moves so rapidly that it appears nearly instantaneously, λ→ 0 and the solution
converges instead to Wagner’s function.
1.2.3 Harmonically oscillating wings
Theodorsen [14] developed a solution for the unsteady lift produced by a wing
which harmonically oscillates. Theodorsen’s work allowed for wing plunge and pitch
and for the deflection of a trailing edge flap. Theodorsen’s solution divides the
force production into circulatory and non-circulatory (intertial) and expresses the
9
unsteady circulatory lift as an amplitude and phase variation of the quasi-steady
circulatory lift. The complex “Theodorsen function,” C(k), presents the gain and
lag of unsteady force production as a function of reduced frequency, k = fc
2U∞ .
As with preceding work, Theodorsen’s analysis required that the flow was
two-dimensional, inviscid, incompressible, and that the motions of the wing were only
small perturbations compared to the mean flow. Garrick [15] extended Theodorsen’s
work by solving for the thrust generated by the same conditions. Isaacs [16] and
Greenberg [17] each extended Theodorsen’s work to include oscillating longitudinal
velocity.
1.2.4 Arbitrary wing motion
von Ka´rma´n and Sears [18] produced an analytical solution for the unsteady
lift caused by arbitrary small-perturbation motion of a two-dimensional thin wing
in a spatially varying gust. The quantification of force production was approached
differently than in previous works; the force production was related to the growth and
movement of vorticity rather than directly to surface pressure. This work combined
and extended the solutions of Wagner, Ku¨ssner, and Theodorsen, and emphasized
the physical significance of the resulting equation. The lift production was separated
into three physically-distinct terms (for each of these terms, the wing chord is 2 units
long and the lift is expressed per unit span):
Quasi-steady lift: The lift corresponding to the enforcement of the Kutta condition.
This quasi-steady thin airfoil theory term is unaffected by the vorticity in the wake
10
of the wing.
L0 = ρUΓ0 (1.9)
where ρ is the fluid density, U is the horizontal velocity of the wing, and Γ0 is the
bound circulation required to achieve tangential velocity at the trailing edge.
Inertial lift: The lift produced as an inertial response to the acceleration of the
surrounding fluid. This is called “apparent mass” or “added mass” and is presented
as:
L1 = ρ
d
dt
∫ 1
−1
γ0(x)xdx (1.10)
where x is the spatial coordinate along the chord of the wing and γ(x) is the chordwise
distribution of vorticity on the wing. As is typical, x is measured in semi-chords
away from the mid-chord of the wing.
Unsteady lift: The vorticity in the wake of the wing influences the vorticity
distribution on the wing and so alters the lift. This is presented as:
L2 = ρU
∫ ∞
1
γ(ξ)√
ξ2 − 1dξ (1.11)
where ξ is the spatial coordinate extending horizontally into the wake and γ(ξ) is
the vorticity distribution in the wake. ξ is also measured as semi-chords behind
mid-chord.
This analytical solution is limited to conditions in which the flow around the
airfoil is two-dimensional, the vertical wing movement and fluid velocity is small
11
enough that the wing and wake can be considered to lie on the x-axis everywhere,
the flow remains attached to the wing, and the Kutta condition is enforced. In
addition to these limitations, the model by von Ka´rma´n and Sears requires knowledge
of the vorticity distribution over the wing and in the wake, or some quantified
relation between wing kinematics and that vorticity. von Ka´rma´n and Sears used
this to recreate the solutions of Wagner, Ku¨ssner, and Theodorsen, for which the
vorticity distributions were solvable; such solutions are not available for separated
flow phenomena. It is the physical interpretations of lift production that will be
useful in the current work rather than the vorticity-impulse equations themselves.
Unsteady force production in cases of separated flow has been addressed more
recently by applying potential flow analysis to shear layers shed from both the
leading and trailing edges. A blade-element model was developed for insect-like
wings in flapping hover by Ansari et al. [4, 19, 20]. A similar model was developed
by Eldredge and Darakananda [21, 22] which applies to general wing motion but
requires empirical tuning. Each of these models lacks a closed-form solution, and
was designed to be implemented by numerical integration.
1.2.5 Indicial theory
Wagner’s function and Ku¨ssner’s function represent indicial responses, which
can be used to find the response of the system to arbitrary forcing using Duhamel’s
integral [23]. This technique approximates arbitrary forcing (wing kinematics or gust
velocity distributions) as a series of independent step changes. The resulting lift
12
from each step gust can be superimposed as long as the system response is a linear
function of the forcing; the small perturbation and small angle assumptions used in
Wagner’s and Ku¨ssner’s theories have already linearized the aerodynamic response.
When taken to the limit of infinite step changes of infinitesimal magnitude,
this approximation becomes the integral of the indicial response and the rate of
change of the forcing conditions. Using Ku¨ssner’s function as an example, the lift
coefficient, Cl, caused by a wing translating at constant velocity, U∞, through an
arbitrary gust field, wg(x), using convective time, s, as above and σ as a bound
variable for integrating to s, is [2]:
Cl(t) =
2pi
U∞
(
wg(0)ψ(s) +
∫ s
0
dwg(σ)
dσ
ψ(s− σ) dσ
)
(1.12)
Here 2pi is the normalized lift curve slope, wg(σ)
U∞ is the linearized flow incidence angle,
and ψ(s) is Ku¨ssner’s function. The first term captures the initial condition, and the
second term integrates the cumulative effects of flow velocity changes for 0 ≤ σ ≤ s.
1.3 Empirical adaptation
More recently, contributors to AVT-202 [24–27] developed a simplified model
for force production by flat plate wings surging from rest at a fixed, high angle of
attack. By significantly limiting the allowable wing kinematics, they were able to
substitute empirical trends for some of the more complex components of analytical
models like that by von Ka´rma´n and Sears.
13
Pitt Ford and Babinsky [28] suggested that flow separation for a thin flat plate
at high incidence results in negligible bound circulation. Instead of creating attached
flow with vorticity contained in a boundary layer, the sharp edges of the plate cause
flow separation which causes the vorticity to collect in leading and trailing edge
vortices. In Babinsky et al. [26], the observed influence of leading edge separation on
transient lift production led the new model to replace the “bound” circulation (from
the Kutta condition in attached flow) with a modeled leading edge vortex. The wake
is further simplified into two point vortices of strength Γ instead of wake sheets with
vorticity distributions. This eliminates the need to solve integrals when determining
the force using the vorticity impulse method. To better represent the observed wake
behavior with only two vortices, these vortices convect relative to the background
fluid — unlike the stationary wake of von Ka´rma´n and Sears. The influence of these
vortices on the unsteady lift is:
Cl, circ. = − 2
U2∞c
[
(uLEV − uTEV) Γ + (xLEV − xTEV) Γ˙
]
(1.13)
The terms inside the brackets are vortex strength, Γ, scaled by the vortices’ relative
velocity, (uLEV − uTEV), and vortex growth, Γ˙, scaled by the vortices’ separation
distance, (xLEV − xTEV).
Estimates of the vortex sizes and trajectories, which were needed to calculate
lift production, were made based on experiments [25]. The relative vortex velocity
14
was approximated as half of the free-stream velocity:
uLEV − uTEV ≈ U∞
2
(1.14)
This approximation only holds in the immediate transient for a surging wing; the
relative vortex velocity will approach zero as each vortex becomes stationary relative
to the background flow. The horizontal separation of the vortices was approximated
as the horizontal projection of the wing chord:
xLEV − xTEV ≈ c cos(α) (1.15)
This separation is constant, which conflicts with the inclusion of non-zero relative
vortex velocity but does not significantly affect the results [26]. The vortex circulation
is then approximated by a scaled modification of Wagner’s function, φ(s). Typically,
a varying velocity profile would be incorporated using indicial theory (see section
1.2.5). This was rejected in favor of a simpler approximation; the gradual nature of
the surging acceleration is reconciled with the impulsive start of Wagner’s function
by scaling the result by the ratio of instantaneous to final velocity:
Γ ≈ Γ∞U(t)
U∞
φ(t) (1.16)
The steady-state value of circulation was approximated using linearized thin-airfoil
15
theory:
ρU∞Γ∞ ≈ (2piα) 1
2
ρU2∞c (1.17)
∴ Γ∞ ≈ piαcU∞ (1.18)
The non-circulatory added-mass lift was approximated using potential flow for an
accelerating flat plate, equation 1.6. All of these approximations together form a
simplified quasi-steady model for lift produced by a surging-from-rest flat plate:
Cl =
−2piαU(t)
U2∞
[
U∞
2
φ(t) + c cos(α) φ˙(t)
]
+
pic
2U2∞
U˙ sin(α) cos(α) (1.19)
The left half of equation 1.19 represents the circulatory force production by vorticity
in the wake of the wing. The right half of the equation represents the non-circulatory
force production by the acceleration of the wing. The simplifications made in the
derivation of this model have made it easy to implement, but have restricted its
applicability to surging-from-rest flat plates at high wing pitch angles.
1.4 Experiments
Unsteady aerodynamic force production from translational acceleration (surge
and pitch) has also been investigated experimentally. Force measurements in cases
with flow separation are particularly interesting, as those results have allowed the
direct comparison to results of predictive models. Measurements (and qualitative
visualizations) of the associated flow velocities have allowed the testing of the validity
of each model’s assumptions, such as verifying attached or separated flow. Existing
16
literature has measured forces and tested models for a range of conditions, including
harmonic oscillations in surge, plunge, and pitch, and surge-from-rest maneuvers.
The current work adds to this by investigating non-oscillating transverse events for
both moving-fluid and moving-wing cases.
1.4.1 Streamwise perturbations
Dickinson and Go¨tz [29] measured unsteady force production for rapid surge-
from-rest maneuvers. Inertial force production was intentionally neglected, but it
was concluded that even post-acceleration the transient lift and drag were higher
than steady-state. Both the transient and steady-state forces were predominantly
chord-normal. Mancini et al. [30] further studied the force production and vortex
characteristics for a surging flat plate during a variety of start-from-rest maneuvers.
The unsteady force production was compared to the sum of two terms, added-
mass inertial forces and a quasi-steady simplification of Wagner’s function. It was
concluded that the leading edge vortex caused the observed transient lift to exceed
the prediction made by the two quasi-steady terms. It was also noted for high-
incidence cases that the recovery time was greater than expected; the unsteady
forces did not return to steady-state values within the measured duration of 14
chord-lengths traveled. Stevens et al. [25] continued to study surging flat plates.
Results were compared from different facilities (and computational methods) to
confirm the presence and influence of shed vortices at high incidence. The vortex
dynamics observed in this work allowed for the simplifications in the semi-empirical
17
model by Babinsky et al. [26].
Much of the experimental literature on gust encounters focuses on longitudinal
gusts, which can be created by varying wind tunnel speed [31–33]. The alignment
of the steady and transient velocities in the longitudinal cases makes them quasi-
one-dimensional, and so they are experimentally and analytically simpler than
the transverse cases. Granlund et al. [33] presented a comparison of oscillating
longitudinal gust experiments using two facilities. One facility was fitted with
rotating vanes in the tunnel to create an oscillating free stream velocity around a
stationary model. The other facility oscillated the model within a steady free stream
flow. It was concluded in that work that the analytical estimation of added-mass
forces was accurate and that subtracting the inertial forces in the moving model case
allowed for the direct comparison of moving-model and moving-fluid results.
1.4.2 Transverse perturbations
More literature exists for transverse unsteadiness caused by plunge maneuvers
rather than fluid gusts. This is particularly common in the context of flapping-wing
MAVs, and so most of the work on this subject has focused on frequency-based
models for wings which harmonically oscillate in plunge (and often also in pitch).
Unsteady force measurements were shown to exceed quasi-steady predictions [34, 35].
The measured forces matched predictions by Theodorsen’s model (at least as closely
as did predictions by quasi-steady or computational models) despite flow separation
[36–38]. Flow visualizations and flow velocity measurements indicated that the
18
unsteady force was strongly affected by the wing-vortex interactions [39–41].
Non-harmonic plunge maneuvers are not covered so thoroughly. Kriegseis et al.
[42] studied both surging and plunging maneuvers in an investigation of the influence
of flow history. The transient velocities in each case were ramp-up maneuvers, which
are commonly studied for surge but not for plunge. Each maneuver was designed
to reach the same steady-state condition from different initial conditions, and it
was concluded that the pre-maneuver presence of a developed boundary layer in the
plunge case had little effect on the unsteady force production.
Transverse gusts are more difficult to experimentally recreate than longitudinal
gusts, and are less commonly addressed in the literature. Kuethe [43, 44] created a
gust encounter by moving a wing over a vertically-oriented open-section wind tunnel.
The test article was mounted at the end of a whirling arm as it would have been for
steady aerodynamics tests. At one location around the circular path of the wing, the
wing passed over a gust generator which allowed for the measurement of transient
forces. The lift produced during the gust encounter was found to be in agreement
with predictions made by the theory of von Ka´rma´n and Sears [18]. Holmes [45]
created an extraordinary wind tunnel with cam-actuated, sinusoidally-waving walls.
The phasing of the cams determined the shape of the wind tunnel interior which
could be configured to create an oscillating longitudinal or transverse gust at the
wing. It was concluded that analytical models were valid for low-amplitude gusts, but
that flow separation at higher amplitudes caused significant disagreement between
predictions and observations. The highest ratio of the gust velocity to the steady
velocity, the “gust ratio,” among these tests was G = 0.24.
19
Wong et al. [46] compared experimental moving model results and
computational moving fluid results for harmonic transverse gusts to investigate
their possible equivalence. Unlike the longitudinal case by Granlund et al. [33], Wong
et al. concluded that inertial effects were not the only difference between the two
arrangements. This was highlighted by observing that the forces in each case were
different even at the moment that each transverse acceleration was zero.
1.5 Present work
Preceding work has incrementally advanced the understanding of unsteady
aerodynamic force production in various related aspects, but none have yet thoroughly
addressed the force production by non-harmonic transverse unsteady events which
are too large to be simplified by small-perturbation assumptions. This document
describes an experimental investigation of transverse gusts and plunging maneuvers
which works towards a closed-form analytical model of force production with flow
separation. This is similar to the approach of Babinsky et al. [26]; the physical
understanding granted by the analytical solution of von Ka´rma´n and Sears will be
adapted based on empirical findings so that the resulting model better represents
the high-amplitude unsteadiness of fluid gusts and plunging wing maneuvers for
limited kinematics. Some time will also be taken to expand on the moving-model to
moving-fluid comparisons of Granlund et al. [33] and Wong et al. [46].
Chapter 2 describes the experimental facility and methods which enabled this
work. The test apparatus, tools for data collection, and methods of data processing
20
are presented to aid in understanding the following results. Chapter 3 presents and
discusses the unsteady force production for transverse gust encounters. Chapter
4 presents the measured forces from various plunging wing maneuvers designed
to be geometrically similar to the gust encounters. It also directly compares the
force production of the moving-model and moving-fluid cases. Finally, chapter 5
summarizes the present work and its significant conclusions, and presents several
suggestions for future work.
21
Chapter 2: Methods
The current work is primarily experimental; its foundation is the measurement
of forces and fluid velocities resulting from deliberately controlled gust encounters
and plunging maneuvers of a flat plate wing. This chapter discusses the towing tank
facility in which these experiments took place, the gust generator added to that tank
for these experiments, the flat plate wing used as the test article, the techniques
used for collecting the data, the parameters defining the tested conditions, and some
general treatment of the collected data.
2.1 Towing tank
Experiments were conducted in the University of Maryland’s water-filled towing
tank, shown in figure 2.1. The tank is 7 m long, 1.5 m wide, and the water in it is
1 m deep. Using water as the working fluid instead of air allowed for higher forces at
the achievable towing speeds and greatly improved the ability to conduct particle
imaging velocimetry (PIV) on highly unsteady and separated flows.
The tank had a steel frame holding glass walls and a glass floor which allowed
optical access for PIV. The frame also carried a four degree-of-freedom motion control
system for maneuvering the test article. The motors were controlled by PID feedback
22
Figure 2.1: The experimental facility, a water tow tank with four degree-of-
freedom motion control.
based on encoder position readings. Three degrees of freedom were used in the
current experiments: streamwise towing and two parallel vertical control rods. The
parallel inputs were converted by a slide-crank mechanism to stream-normal motion
and angle of attack.
2.2 Test article
The test article was the simplest form of lifting surface, a flat plate wing. In
these experiments the flat plate had a 5 cm chord, 20 cm span, and 1.6 mm thickness.
This makes the aspect ratio 4 and the thickness-to-chord-ratio 3%. This wing was
designed to match that of previous experiments by AVT-149 [6].
Two wings of this design were created using different materials. The first of
23
Figure 2.2: CAD rendering of the glass flat plate wing attached to the force
balance and the maneuvering apparatus.
these was made of brass. This metal wing was machined using a computer numerical
control (CNC) mill to ensure accurate sizing and edge characteristics. Brass was
selected because it is more corrosion resistant in water than aluminum and lighter
than stainless steel. The second wing was made of glass to allow for PIV illumination
without a shadowed region. Glass was selected over transparent plastics for high
bending stiffness. The manufacturing process for the glass wing did not allow the
same precision (for square edges in particular), but no difference in results was
observed between the two wings. The wing and the apparatus for maneuvering it
are shown in figures 2.2 and 2.3.
The static force coefficients of the flat plate wing are shown in figure 2.4. The
24
Figure 2.3: The brass flat plate wing and the maneuvering apparatus
prepared for installation. Black paint reduces PIV laser reflections.
10 20 30 40 50 60 70 80 90 100
-0.5
0
0.5
1
1.5
2
Figure 2.4: Static force coefficients measured for the flat plate wing at
Re = 20,000.
forces are predominantly plate-normal, although a small plate-tangent force is present
even at α = 0°. The peak lift coefficient is CL = 0.95, produced at α = 23°. The
plate-normal force coefficient has a local maximum at α = 25° of CN = 1.1, and a
peak value of CN = 1.39 at α = 90°. This is also the peak drag value and angle.
These measurements agree with those published by Ortiz et al. [47] if the difference
in wing aspect ratio is properly accounted for.
25
Figure 2.5: The plumbing and nozzle array of the gust generation system.
The arrows indicate the flow direction.
2.3 Gust generator
The fluid gust created for these experiments was a vertically-oriented,
approximately planar water jet. A pool circulation pump moved 60 gallons of
water per minute through a recirculating system with an outlet manifold made of 30
cylindrical nozzles, shown in figure 2.5. The nozzles had 1
4
in. inner diameter and
were spaced 1
2
in. center-to-center. The conical jets that form from each nozzle merge
into one planar jet above the nozzle array. This jet passes through a size 28 steel
mesh (0.7 mm wire spacing, 52% open area) which was used as a low-profile flow
26
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Streamwise width (m)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Figure 2.6: Streamwise gust velocity distribution extracted from PIV with
and without the flow-straightening screen. The shaded region around each
curve shows the standard deviation of the velocity over time. The gust
profile was approximated as V (x) = 0.34 sin
(
pix
0.16
)2
for 0 ≤ x ≤ 0.16.
straightener to remove the streamwise waving of the planar jet. Figure 2.6 shows
the velocity profile of the gust with and without the screen. The screen reduces the
variations over time (the shaded region in figure 2.6 shows one standard deviation
in gust velocity measurements) but also reduces the time-averaged peak velocity of
the gust. The jet continues vertically through the water tank to inlet pipes at the
top surface, which draw in the water for the pump. This forms a closed loop system
with a “free jet” (conceptually similar to Kuethe’s design in section 1.4). Away from
the spanwise ends of the nozzle array, the gust can be considered approximately
two-dimensional and can be characterized by a streamwise distribution of transverse
velocity. The gust system manifold is 37 cm wide, which covers 1.85 times the
wingspan. This leaves 1.7 chord-lengths between each wingtip and the spanwise ends
of the manifold, so the wing is within the two-dimensional flow region of the gust.
27
The uniformity of the spanwise velocity distribution was confirmed qualitatively with
dye flow visualization. At the height above the nozzles used for all tests, 15 cm, this
gust formed a canonical sine-squared velocity profile with a span of 16 cm and a peak
velocity of 0.34 m
s
. This best-fit profile can be compared to the mean measured gust
velocity profile in figure 2.6.
2.4 Gust encounter kinematics and parameter matrix
Force data was collected as the flat plate wing was towed at constant velocity
and constant pitch angle through the transverse gust. The size of the tow tank
limits the total towing distance to approximately 6 m. Figure 2.7 shows the tested
kinematics, which in detail were: 6 chord-lengths of constant acceleration from rest
to full towing speed, 25.9 chord-lengths of constant velocity before reaching the
gust, 3.2 chord-lengths in the gust, 60.9 chord-lengths of constant velocity after
the gust (90 total chord-lengths, 4.5 m, of constant velocity towing), and finally 6
chord-lengths of constant acceleration to return to rest.
Those kinematics were repeated for several values of towing velocity and angle
of attack. The tested gust encounter cases were limited by the force measurement
sensitivity for low speeds and by the motor capabilities for high speeds. Tests were
conducted at freestream Reynolds numbers of Re ≡ U∞c
ν
= 10,000, 20,000, and
40,000. These three cases can be described by their corresponding wing velocities of
U∞ = 0.2, 0.4, and 0.8 ms , or by the associated peak gust ratios of G = 1.68, 0.84,
and 0.42. The pitch angle of the wing was also varied from −4° to 45°. The primary
28
-50 0 50 100
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Towing velocity
Gust velocity
Figure 2.7: The horizontal towing velocity of the wing and the vertical gust
velocity at the leading edge of the wing over the course of a Re = 20,000
case.
cases tested at each Reynolds number were 5°, 10°, 15°, 20°, and 45°; these cases are
emphasized with bold pitch angles in table 2.1, and were repeated between 4 and
10 times. A distribution of additional low angle cases were tested; these cases were
measured only once or twice. 140 total measurements were made between the 30
cases.
The irregular cases in the test matrix were caused by hardware limitations
of the motion control system. The system was able to set a specified wing pitch
angle by feedback control, but this could allow the wing to change angle of attack
briefly during the gust encounter (as it was perturbed by the gust forcing). This was
avoided by increasing the input threshold for motion of the slide-crank mechanism
which was controlled by torque on the pivoting and sliding pins. This also worked
against the motion control system, which occasionally became fixed at an angle other
29
Table 2.1: Cases tested transverse gust encounters characterized by gust
ratio and pitch angle.
Reynolds Gust ratio, Pitch angles,
number G α (o)
10,000 1.68 2, 3, 4, 5, 6, 7, 10, 15, 20, 45
20,000 0.84 2, 4, 5, 6, 10, 15, 20, 45
40,000 0.42 -4, -1, 0, 1, 3, 5, 6, 7, 10, 15, 20, 45
than the prescribed angle. Instead of discarding the collected force data in those
cases, the achieved angles were recorded and new repetitions were measured until
the prescribed angles were accurately reached for enough measurements. This issue
was resolved by further PID tuning before the more recent tests described below.
2.5 Plunge maneuver kinematics and parameter matrix
The maneuver for all cases in this study was the sine-squared gust, also known
as the one-minus-cosine gust due to the identity 2 sin(θ)2 = 1− cos(2θ). This profile
was chosen to match the canonical approximation of fluid gust encounters [1] to
facilitate comparison of this data to transverse gust encounter data. The plunge
velocity of the wing during the maneuver was:
V (t) = Vp sin
(
pit
T
)2
for 0 ≤ t ≤ T (2.1)
where Vp is the peak plunge velocity and T is the maneuver duration. This is
shown in figure 2.8. By integration, the vertical distance traveled during the gust
is h = 1
2
VpT . The wing traveled at constant horizontal velocity, U∞, therefore
30
Time
V
el
oc
ity
Figure 2.8: Plunge velocity profile for h
w
= 1.
the horizontal distance covered during the gust was w = U∞T . The duration and
chord-normalized size of the maneuver (T , w
c
, and h
c
) are the controlled variables for
this experiment.
This study measured forces for 135 different plunge maneuvers, which primarily
varied in height and width traveled by the wing during the maneuver. There were
72 unique combinations of height and width, each ranging from 2.5 cm to 30 cm, or
0.5c to 6c. The path of the wing during three of these plunge maneuvers and the
endpoints of each maneuver are plotted in figure 2.9. The plunge begins at the top
left corner of the plot and ends at the associated endpoint marker for each case.
Two plunge speeds were tested. Each of these 72 maneuver sizes was tested at
a duration set so that the peak total velocity was Umax = 0.75
m
s
by:
T =
√
(2h)2 + (w)2
Umax
(2.2)
31
0 1 2 3 4 5 6
0
1
2
3
4
5
6
Figure 2.9: Plunge profile for 3 of the 72 maneuver sizes in the test matrix.
The marked grid of points represent the end location of each tested maneuver.
Cases with integer-valued sizes, plotted as circles in figure 2.9, were tested again at
Umax = 0.25
m
s
. Nine of the original cases were repeated at angles of attack above
zero: 10°, 20°, and 45°. The full test matrix is presented in table 2.2.
The Reynolds numbers based on the peak total velocity, Umax, are 37,500 and
12,500. The Reynolds numbers based only on the horizontal velocity, U∞, vary
between 1,038 (w = 1c, h = 6c, Umax = 0.25
m
s
) and 36,895 (w = 5.5c, h = 0.5c,
Umax = 0.75
m
s
).
The positions of the motors were recorded during each test at 32 Hz. Figure
2.10 shows the plunge kinematics as prescribed and performed over three repetitions
32
Table 2.2: Experimental test matrix characterized by size and speed of the
plunge maneuver. Colors correspond to cases in figure 2.9
w
c
h
c
Umax
(
m
s
)
α
0.5 0.5, 1.5, 2.5, 3.5, 4.5, 5.5 0.75 0°
1 1, 2, 3, 4, 5, 6 0.25, 0.75 0°
1.5 0.5, 1.5, 2.5, 3.5, 4.5, 5.5 0.75 0°
2 1, 2, 3, 4, 5, 6 0.25, 0.75 0°
2.5 0.5, 1.5, 2.5, 3.5, 4.5, 5.5 0.75 0°
3 1, 2, 3, 4, 5, 6 0.25, 0.75 0°
3.5 0.5, 1.5, 2.5, 3.5, 4.5, 5.5 0.75 0°
4 1, 2, 3, 4, 5, 6 0.25, 0.75 0°
4.5 0.5, 1.5, 2.5, 3.5, 4.5, 5.5 0.75 0°
5 1, 2, 3, 4, 5, 6 0.25, 0.75 0°
5.5 0.5, 1.5, 2.5, 3.5, 4.5, 5.5 0.75 0°
6 1, 2, 3, 4, 5, 6 0.25, 0.75 0°
1 1, 3, 6 0.75 10°, 20°, 45°
3 1, 3, 6 0.75 10°, 20°, 45°
6 1, 3, 6 0.75 10°, 20°, 45°
for one of the most aggressive plunge maneuvers, w = 1c, h = 1c. Minor overshoot
of 0.04 m
s
(6% of the peak velocity) was recorded at the end of the maneuver. The
motors follow the prescribed kinematics even more closely in most cases — the
average overshoot is 0.6% of the plunge velocity.
33
-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
-0.8
-0.6
-0.4
-0.2
0
-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3
-0.8
-0.6
-0.4
-0.2
0
Prescribed motion
rep. 1
rep. 2
rep. 3
Figure 2.10: Plunge motor velocities as prescribed compared to the performed
motor motions over three repetitions of an aggressive plunge maneuver
(w = 1c, h = 1c).
2.6 Force measurements
The forces produced by the gust encounters and plunging maneuvers were
measured using a six degree-of-freedom force and torque balance (ATI Mini-40)
sampled at 1 kHz. The force balance was located between the slide-crank pitch
mechanism and the sting, as is visible in figure 2.2. Internal strain readings were
converted to wing-fixed forces and moments using the manufacturer’s calibration.
Figure 2.11 shows lift coefficient measurements for eight repetitions of an example
gust encounter case. The gust encounter takes place during the shaded portion of
the x -axis.
34
-5 0 5 10 15 20 25 30 35 40 45
0
0.5
1
1.5
2
2.5
rep. 1
rep. 2
rep. 3
rep. 4
rep. 5
rep. 6
rep. 7
rep. 8
Mean
Smooth
Figure 2.11: Lift measurements from individual repetitions, ensemble average,
and post-processing for one gust encounter case. α = 45° and G = 0.84
(Re = 20,000).
The data packets were transmitted from the moving tow tank carriage to the
stationary control computer by a dedicated wireless network. The data transfer took
place real-time and was limited to User Datagram Protocol (UDP) to achieve that
speed. That protocol is simple and efficient, but is unconfirmed, so any individual
data packets which were transmitted incorrectly or incompletely were unrecoverable.
Data was recorded at a high rate to ensure that enough temporal measurement
resolution was achieved even when some data points were lost. If a single run of
data collection (one repetition of one case) lost more than 5% of the data packets, it
was discarded and restarted.
Mechanical vibrations created measurement noise that limited the lowest viable
35
Reynolds number. The standard deviation of repeated measurements was 0.1 N,
which corresponds to a measurement uncertainty of ±2% of the mean forces measured
for cases at Re = 40,000, ±7% for cases at Re = 20,000, and ±30% for cases at
Re = 10,000.
The guide rails and magnetic propulsion of the towing carriage were not
perfectly smooth and sometimes introduced substantial transient vibrations to the
measured forces, as is visible in figure 2.11 at t∗ = 16. For all plunge maneuver
force data the starting location of the test is shifted by a random movement of
up to ±100 mm so that the fixed location of the track bumps appear as varying
locations in the experiment. This minimizes the effects of the track unevenness on
the ensemble-averaged data. However, the location of the gust system in the tank
was not variable, so the starting location of the gust encounter tests could not be
varied.
2.6.1 Post-processing
Force measurements for this experiment were repeated and the results were
ensemble-averaged. However, the force data collection device and wing kinematics
control computer operated on separate internal clocks, so the measurements had
to be synchronized to the kinematics timeline before averaging or further analysis
was possible. The kinematics and force measurements were triggered to begin
simultaneously, so each could be simply considered relative to the start time. This
was found to introduce an error of up to 0.1 s, which significantly affected the averaged
36
0 5 10 15 20 25 30
0
0.5
1
1.5
2
2.5
3
3.5
4
Measured
Calculated
Aligned
Figure 2.12: Force measurements were aligned to kinematics by correlation
of water pressure and transducer depth.
results. Instead, the measured side-force was cross-correlated with the expected
side-force — based on the transducer’s depth in the water and the total velocity
— to find the best alignment. This alignment can be seen in figure 2.12, which
shows the side-force for one of the plunge maneuver repetitions (for which h = 1
and w = 1). The experimental hardware is symmetric, so the measured side-force
is only affected by the water pressure at the transducer’s measurement face. The
changes in pressure can be approximated as a function of transducer depth and speed,
∆P = ρg∆h + 1
2
ρ (∆Utotal)
2. The initialization of the test article to the starting
position (t < 6 in figure 2.12), the plunge maneuver (t ≈ 17.5), and the return of
the test article to rest (t > 23) each provide significant pressure changes with which
the kinematics and force measurements can be synchronized. In this example, the
37
force measurements timing (in blue) began 0.26 seconds after the kinematics timing
began (from which the red line was estimated). This separation was identified by
normalized cross-correlation of these quantities; the appropriately aligned results are
shown in green.
The data alignment process identified the true time-stamp for each measurement
but each repetition had a slightly de-synchronized sample timing, so the ensemble-
averaged data had to be created based on linearly-interpolated points. This was
acceptable because of the high sample rate; the maximum interpolation “distance”
was 0.5 ms and the unsteady events being studied lasted at minimum 8 ms and on
average 500 ms. Measurement points which were lost by the UDP transfer were not
used in the ensemble average; individual data points for which this occurred were
therefore averaged over one or two fewer repetitions than the overall force history.
Measurement noise and mechanical vibrations created visible noise in the
measured force data. For the remainder of this document, presented force data has
been filtered using weighted quadratic local regression (“loess” in Matlab). This
particular filter reduces noise effectively but also preserves the peaks measured
during these unsteady events. Basic low-pass frequency filters were considered as an
alternative, but were found to either have too little effect or to truncate the peak
forces depending on the cut-off frequency. The span of the loess filter was set based
on convective time so that the filter covered t∗ = 2 in the gust encounter cases and
t∗ = 0.2w in the plunge maneuver cases. It can be seen in figure 2.11 that this did
not truncate the peak force or otherwise distort the force history.
38
-0.5 0 0.5 1 1.5
-10
-5
0
5
10
15
Full experiment
Water removed
Wing removed
Figure 2.13: Measured lift during a plunge maneuver repeated after draining
the tow tank water and again after removing the test article.
2.6.2 Dynamic calibration
The mass of the wing, sting, and parts of the force balance contribute to the
measured inertial loads during the plunge maneuver cases. To isolate the aerodynamic
contributions to those forces, a dynamic calibration procedure was performed on 13
of the plunge test cases by repeating those measurements in air instead of water,
and then repeating each of those cases again with the wing and sting removed.
By comparing the forces measured under those three conditions, the mass of each
component could be verified and the inertial load of the force transducer itself could
be identified. An example of these force results is shown in figure 2.13.
The mass of the glass wing was 38 g, the brass wing was 126 g, the sting was
39
88 g, and the part of the force balance on the measurement side of the strain gages
was determined to be 120 g. The calibration tests in air do experience added-mass
forces, but they are proportional to the fluid density, and so are only 1.2% of the
force experienced in water. For the remainder of this document, the inertial forces
caused by the mass of the test hardware have been removed from the presented
measurements (this is only relevant in the plunge maneuver cases).
Dynamic calibration is often also used to identify the dynamic response of
the measurement equipment. However, in this experiment the effective frequency
of the plunge maneuver was at most 0.5% of the force balance’s resonant frequency
of 3.2 kHz. Therefore, the harmonic response of the force balance did not affect the
measurements and was not further quantified here.
2.6.3 Lift normalization
In cases of high gust ratio, G, or high plunge aspect ratio, h
w
, the majority of
the total relative fluid velocity mid-event is contributed by the unsteady vertical
velocity component. In these cases the typical force normalization by freestream
dynamic pressure leads to unreasonably high force coefficients. The gust ratios in this
experiment did not reach high enough values to cause difficulty in comparisons, but
several of the plunge cases did. The left axis of figure 2.14 shows lift coefficients for
six plunge maneuvers, but most of the data is indiscernible because of the excessive
magnitude of the normalized lift in the w = 1c, h = 6c case. Because of this, the
measured lift in each plunge case is also presented as normalized by the mean kinetic
40
0 1 2 3 4 5 6
-100
-50
0
50
100
150
200
250
300
350
0 1 2 3 4 5 6
-2
-1
0
1
2
3
4
5
6
7
Figure 2.14: Lift normalization by freestream dynamic pressure (left) and by
mean specific kinetic energy (right). The exaggerated magnitude of forces
for the high plunge aspect ratio case on the left diminishes the differences
between the lower aspect ratio cases. The alternate normalization removes
this effect, making all cases visibly comparable.
energy per unit mass during the maneuver [48]. To see the difference, take the typical
definition of lift coefficient,
CL =
L
1
2
U2∞ρS
, (2.3)
which exaggerates the lift produced by maneuvers with high velocity ratios since
the normalizing factor, U∞, is only a small component of the total velocity. If the
dynamic pressure part of the equation is replaced by a term which incorporates the
unsteady vertical velocity magnitude then the normalizing factor better reflects the
41
total velocity of the maneuver. The mean specific kinetic energy is defined as:
K¯E
m
=
1
T
∫ T
0
1
2
(
U2∞ + V (t)
2) dt (2.4)
For the sine-squared plunge profile studied here (equation 2.1), the new normalization
of lift coefficient becomes:
Lnorm =
L
1
2
(
U2∞ +
3
8
V 2p
)
ρS
(2.5)
∴ Lnorm =
CL
1 + 3
2
(
h
w
)2 (2.6)
This new normalization allows for the direct comparison of the varying plunge
maneuvers, as seen in the right axis of figure 2.14. This scaling reduces the magnitude
of the normalized lift for the most aggressive plunge in the current data (w = 0.5c,
h = 5.5c) by a factor of nearly 200, which greatly improves the quantitative and
visual comparisons of the aggressive and gentle plunging force data. In chapter 4 the
forces will be presented using both normalizations for comparison.
2.7 Flow velocity measurements
Particle imaging velocimetry (PIV) was used to measure the flow around the
wing. A Phantom v641 camera recorded 4 MP images (2560× 1600 pixels) at 1 kHz
as it was towed along with the wing. The water was seeded with hollow glass spheres
of 37µm diameter which were illuminated by a Litron LDY300 pulsed Nd:YLF laser.
42
Figure 2.15: Diagram of PIV measurement setup. The camera and laser
optics are suspended from the towing carriage outside of the tank so that
each travels along with the wing.
The beam was spread into a thin sheet by a Powell lens. The measurement plane
was aligned with the towing velocity and the gust velocity, and intersected the wing
one chord-length from the centerline — and, due to the aspect ratio of four, also
one chord-length from the near-camera wing tip. The laser sheet optics were also
towed on the carriage so that the illuminated region traveled with the wing. This
arrangement is shown in figure 2.15. The laser sheet passed through the free water
surface inside of an acrylic carrier block to avoid distortion of the laser sheet by
refraction. The field of view, shown in figure 2.16, was 25× 15 cm. This resulted in a
spatial resolution of 0.1 mm per pixel or 500 pixels per chord-length. The normalized
temporal resolution was 150 images per convective time.
Only one PIV case is discussed in the current work1. This case was recorded
using a towing velocity set equal to the characterized gust velocity, U∞ = 0.34 ms .
1During subsequent PIV measurements, it was observed that the generated gust velocity declined
over several days. The water pump system failed and was found to be unrepairable.
43
Figure 2.16: The field of view for PIV measurements. In this case, α = 45°.
The wing is centered in the image and is visible by the laser illumination at
its surface (even glass reflects a little of the laser). The force transducer is
hidden behind the black foil disc between the sting and pitch mechanism,
which are visible in the lower right of the image.
The wing was set at α = 45° and was centered in the field of view.
The captured images were analyzed in sequence using DaVis software, which
ran multi-pass cross-correlation to measure the seed particle displacements. The
interrogation regions were 32× 32 pixel windows for the first coarse pass and 16× 16
pixels for the second pass. The second pass used a circular weighting to remove
diagonal bias. Each pass used 50% interrogation region overlap to maximize vector
resolution without overly sacrificing measurement independence. The final vector
spacing was 0.75 mm, or 67 vectors per chord-length. The resulting vector fields were
post-processed using local-median outlier removal (7% of vectors were replaced) and
no smoothing.
The velocity fields were used first as flow visualization to observe the vortex
44
dynamics during the transverse gust encounters. Flow vorticity was then calculated
to aid in identifying vortices and to investigate quantitative correlation between
near-wing vorticity and transient force production.
2.8 Chapter summary
A water-filled towing tank was outfitted with a recirculating gust generation
system to experimentally investigate unsteady force production by transverse gust
encounters and plunge maneuvers. Forces were measured using a six degree-of-
freedom force and torque balance and flow velocities were measured using PIV towed
along in the wing reference frame. Gust encounters were recorded for various towing
velocities for which 10,000 ≤ Re ≤ 40,000 and wing pitch angles of −4° ≤ α ≤ 45°.
Forces and flow velocities from gust encounters are presented in chapter 3. Plunge
maneuvers which followed the canonical sine-squared gust velocity profile were also
investigated. Plunge maneuvers were varied in the distance traveled in the streamwise
and stream-normal directions during the maneuver, the duration of the maneuver,
and wing pitch angles between 0° ≤ α ≤ 45°. Freestream Reynolds numbers for
plunge maneuvers were 1,038 ≤ Re ≤ 37,500. Forces from plunge maneuvers are
presented in chapter 4.
45
Chapter 3: Force production in transverse gust encounters
This chapter will present the force and flow measurements made during sine-
squared profile transverse gust encounters of a flat plate wing. These observations
aid in identifying and quantifying mechanisms of unsteady aerodynamic force
production. Transverse gust encounters are investigated for gust velocity ratios
between G = Vg
U∞ = 0.42 and 1.86, and for wing pitch angles between α = −4° and
45°. The details of the tested parameter space are presented in section 2.4. The
analysis begins with a qualitative description of the recorded force histories and a
connection between those forces and the flow structures observed in the measured
velocity fields. Several force prediction models are compared to the measured force
histories. Force predictions made by these models are further compared to measured
forces by extracting peak forces from each gust encounter as characteristic values.
3.1 Gust encounter force history features
The effects of the gust encounter on the force coefficients of the flat plate wing
are visible in figure 3.1. The x -axis of this figure is convective time, t∗ = tU∞
c
, the
number of chord-lengths traveled by the wing since its leading edge entered the gust.
In the case displayed in figure 3.1, the angle of attack is 45° and the gust ratio is
46
0 10 20 30 40
 
0
0.5
1
1.5
2
2.5
0 10 20 30 40
0
0.5
1
1.5
2
2.5
0 10 20 30 40
 
0
0.5
1
1.5
2
2.5
0 10 20 30 40
 
0
0.5
1
1.5
2
2.5
Figure 3.1: Force coefficient history for gust encounter of flat plate at 45°
and a gust velocity ratio of 0.84. The wing and gust overlap during the
shaded portion of the x -axis; the leading edge of the wing enters the gust at
the left edge of this region, t∗ = 0, and the trailing edge of the wing exits
the gust at the right edge of the region, in this case t∗ = 3.88. The forces
are oriented and normalized relative to the towing velocity of the wing. The
circles are for reference in figure 3.2.
0.84. The forces are normalized by the towing dynamic pressure. The lift and drag
decomposition is performed relative to the constant horizontal towing velocity. There
are significant plate-normal forces, but the tangential forces are nearly constantly
zero as is expected for this thin, flat wing at the tested Reynolds numbers. Because
the wing is thin and flat, the forces at the narrow edges can be neglected. This
means that the forces are approximated as acting only on the top and bottom wing
surfaces, so the pressure forces on the wing are entirely plate-normal and skin-friction
forces are entirely plate-tangent. The magnitude of the measured plate-normal forces
47
as compared to the plate-tangent forces indicates that pressure is the dominant
mechanism of force production here rather than skin friction.
As the wing enters the gust, the total local velocity increases and the relative
angle of incidence between the wing and the total fluid velocity increases. Both
of these effects tend to increase the force production on the flat plate (until the
incidence angle increases beyond 90°). In the example in figure 3.1, the normal-force
coefficient increases from 1.2 before the gust is reached to a peak of 2.2 during the
gust encounter. The relative increase in peak force production, 83%, was slightly
greater than the relative increase in peak dynamic pressure, 71%.
As the wing exits the gust, the force production drops below the steady value
momentarily. This event is not adequately explained by the variations of total
velocity and incidence angle, each of which return to the steady towing values
without overshoot. The local minimum in forces observed here is the first clear
indication that the force production by this gust is significantly affected by flow
unsteadiness.
After the wing exits the gust, the force production rises again above the steady
values, although not as high as was observed while the wing was in the gust. The
nominal flow conditions at this time are the same as at all other points outside of
the gust, but the forces are temporarily still affected by the gust encounter. Because
the effects of the gust on force production last longer than the gust encounter itself,
only a force model which incorporates the flow history can properly capture the gust
recovery behavior.
48
3.1.1 PIV flow velocity measurements
Figure 3.2 shows sequential velocity and vorticity fields captured by particle
imaging velocimetry (PIV) during a similar gust encounter to the example case
described above. For the PIV data collection, the wing was slowed to the measured
gust velocity, so the gust ratio is changed slightly to G = 1.0. The flow features
associated with the force history displayed in figure 3.1, for which G = 0.84, are
expected to be qualitatively the same as observed in the PIV results. The circles in
figure 3.1 highlight the associated times of each of the frames in figure 3.2.
Figure 3.2a shows the wake of the wing during steady towing before the wing
reaches the gust. The flow is fully separated and the wake extends horizontally.
Periodic vortex shedding occurs during the steady towing, but compared to the
transient effects of the gust the forces at this moment can be considered steady.
Figure 3.2b shows the flow around the wing one chord-length after entering the
gust. The rapid change in flow magnitude and direction causes vortices to form and
shed from the leading edge of the wing. The nominal gust velocity at the location of
the leading edge is 0.23 m
s
, 69% of the peak velocity. At the same time the velocity at
the trailing edge of the wing is only 0.03 m
s
, 8% of the peak velocity. At this moment,
the forces on the wing are approaching their peak. The peak measured force is
reached at t∗ = 1.6 — exactly when the leading edge reaches the peak gust velocity.
This suggests that the force production is closely tied to the flow conditions at the
leading edge of the wing rather than another chord-wise location or a chord-averaged
characterization.
49
The local minimum in forces occurs during the time captured by figure 3.2c,
t∗ = 4. The gust encounter has passed, so the nominal gust velocities at both the
leading and trailing edge of the wing are 0 m
s
. A large trailing edge vortex, which is
visible in this frame, forms as the gust recovery process begins. Over the previous
two chord-lengths traveled, the velocity gradient across the trailing edge of the
wing caused exceptionally high positive vorticity production. This concentration
of positive vorticity at the suction surface of the wing causes the concurrent lift
deficiency.
Similarly, figure 3.2d shows a leading edge vortex over the wing which causes
the lift to increase to the secondary peak seen at t∗ ≈ 7 in figure 3.1. This moment
is even further removed from the gust encounter, and yet significant unsteady force
production still occurs. The large trailing edge vortex in figure 3.2c causes the
leading edge shear layer, the line of negative vorticity visible in figures 3.2a and 3.2b,
to move downward; its gradual return to horizontal begins with the formation of the
large leading edge vortex seen here.
Figures 3.2e and 3.2f look increasingly similar to the steady condition seen in
figure 3.2a, but only after 20 more chord-lengths traveled do the forces return to
their steady values. It appears that the alternating vortex shedding from the leading
and trailing edges gradually decreases in vortex strength from the maximum seen in
figures 3.2c and 3.2d to the steady shedding present in figure 3.2a.
The influence of near-wing vorticity can also be evaluated quantitatively. Figure
3.3 compares the lift history from figure 3.1 to the positive, negative, and net local
vorticity contained in the PIV data of figure 3.2. The quantification of vorticity used
50
(a) t∗ = −2 (b) t∗ = 1
(c) t∗ = 4 (d) t∗ = 7
(e) t∗ = 10 (f) t∗ = 13
Figure 3.2: PIV results of the gust encounter at α = 45° and G = 1.0. The
central rectangle indicates the wing position, the background color indicates
vorticity, and the curve along the bottom indicates the vertical velocity
profile of the gust. Most vectors were omitted for visual clarity; only 1.2%
are shown here (one in nine vectors were plotted in each direction).
51
-2 0 2 4 6 8 10 12
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Figure 3.3: Comparison of force history from figure 3.1 to local vorticity for
a gust encounter at α = 45° and G = 1.0.
in this comparison is an inverse-distance weighted average, which emphasizes the
vorticity in the area close to the wing. This is accomplished by scaling each vorticity
measurement by the radial distance from the mid-chord of the wing:
γ¯ =
∑ γ√
(x−x0)2+(y−y0)2∑
1√
(x−x0)2+(y−y0)2
(3.1)
This weighted average diverges as the radial distance approaches zero. That does
not effect the results in the current work because the PIV data is masked at the
wing surface.
The near-wing vorticity corresponds closely with the simultaneous force
production. The negative circulation is associated with the leading edge vortex
52
and lift augmentation, while the positive circulation is associated with the trailing
edge vortex and lift reduction. Both the lift and the negative local vorticity increase
from t∗ ≈ 0 to t∗ ≈ 2, and then decrease to t∗ ≈ 4. The positive local vorticity
increases and decreases similarly, but with a delay of t∗ ≈ 2. The peak in positive
local vorticity occurs at the same time as the minimum in force production. The net
local vorticity at this moment is positive (note that −γnet is displayed in figure 3.3)
and the lift is below the steady-state value. After this, a leading edge vortex forms
which causes the secondary lift peak. These events correspond to the flow fields in
frames 3.2c and 3.2d.
3.1.2 Trends with respect to gust parameters
In order to better understand the effects of gust encounter characteristics on
the resulting force production, measured forces were compared for cases which vary
in only one gust parameter at a time. Figure 3.4 shows force coefficient histories for
three example cases with varied pitch angles of α = 5°, 20°, and 45° at the same gust
ratio, G = 0.84. Note that the wing pitch angle, α, is the constant angle of the wing
chord relative to horizontal. The relative fluid velocity is θ = tan
(
V(t∗)
U∞
)−1
, which
varies over the gust encounter and in this case peaks at 40°. The total incidence
angle of the fluid on the wing is the sum of these components, α + θ. The peak
normal force coefficient increases with pitch angle to α = 20°, but then decreases
as the pitch angle increases from 20° to 45°. The variation in peak normal force
coefficient between these three cases, CN ≈ 2 to 2.5, is notably smaller than the
53
-5 0 5 10 15
 
0
0.5
1
1.5
2
2.5
 5o
20o
45o
-5 0 5 10 15
 
0
0.5
1
1.5
2
2.5
-5 0 5 10 15
0
0.5
1
1.5
2
2.5
-5 0 5 10 15
 
0
0.5
1
1.5
2
2.5
Figure 3.4: Force coefficient history for gust encounters at varied pitch
angles. The gust ratio was 0.84 in each case. The wing is in the gust during
the shaded portion of the x -axis. The forces are oriented and normalized
relative to the towing velocity of the wing.
variation in steady normal force coefficients, CN ≈ 0.4 to 1.2. The magnitudes of
the force oscillations during the gust recovery — the local minimum and following
secondary maximum in force production as the wing exits the gust — are reduced at
lower wing pitch angles.
Figure 3.5 shows force histories for example cases with varied gust ratios of
G = 0.42, 0.84, and 1.68, but the same pitch angle, α = 45°. The peak force
coefficients during the gust encounter increase with gust ratio (for all other tested
pitch angles as well), which was expected since the forces are normalized by the
steady flow velocity so higher gust ratios experience higher normalized dynamic
54
-5 0 5 10 15
 
0
1
2
3
0.42
0.84
1.68
-5 0 5 10 15
 
0
1
2
3
-5 0 5 10 15
0
1
2
3
-5 0 5 10 15
 
0
1
2
3
Figure 3.5: Force coefficient history for gust encounters at varied gust ratio.
The plate was set at 45° in each case. The wing is in the gust during the
shaded portion of the x -axis. The forces are oriented and normalized relative
to the towing velocity of the wing.
pressures:
qpeak
q∞
=
1
2
ρ
(
U2∞ + V
2
g
)
1
2
ρU2∞
= 1 +G2 (3.2)
The unaccounted-for additional dynamic pressure causes the force coefficients to
increase with gust ratio. The magnitudes of the force oscillations during the gust
recovery also increase with increasing gust ratio.
The timing of the force oscillations as the wing exits the gust were consistently
correlated with the trailing edge position. The slight misalignment of those oscillations
in figure 3.4 (most visible at the force minimum) is an effect of the reduced projected
chord of the wing due to the pitch angle. The leading edge of the wing exits the
55
gust at the same convective time in each case (t∗ = 3.18). The time at which the
trailing edge exits the gust depends on the pitch angle: t∗ = 4.18 at α = 0°, t∗ = 4.17
at α = 5°, t∗ = 4.12 at α = 20°, and t∗ = 3.88 at α = 45°. The primary example
case is α = 45°, so the boundary of the shaded area in each figure is t∗ = 3.88. The
connection between trailing edge location and force oscillation timing reinforces the
notion that the trailing edge vortex dynamics are the cause of the lift deficiency.
3.2 Force history predictions
The preceding results were used to guide the adaptation of existing models of
aerodynamic force production to the current problem of large-amplitude transverse
gust encounters. The measured force histories were compared to several theoretical
models in order to identify which models accurately predict the force production for
segments of the studied parameter space. The captured or missing elements of each
model also helped to identify the dominant mechanisms of force production during
the gust encounters. The models considered here were a quasi-steady empirical
look-up table and three adaptations of Duhamel’s convolution integral applied to
Ku¨ssner’s function. One adaptation was based on the empirical data, and two were
based on thin-airfoil theory.
The simplest tool for predicting unsteady forces is an empirical quasi-steady
model. A quasi-steady model bases instantaneous force predictions on instantaneous
conditions, and does not incorporate the flow history in any way. This limits the
model to predictions of forces during the gust encounter; the nominal instantaneous
56
conditions after the gust are unchanging so the forces predicted by this method do
not change outside of the gust either. This model is empirical because the force
coefficients at any time are based on the associated force coefficients measured during
matching steady conditions. This is represented by the following equations, in which
a tilde is used to indicate static data from matching flow incidence angles. The
relative flow angle in the case of the gust encounter varies across the wing chord, so
the static data is matched to a single location, the leading edge:
CN(t) = C˜N(α+θ(t))
(
1+
V (t)2
U2∞
)
(3.3)
CT (t) = C˜T (α+θ(t))
(
1+
V (t)2
U2∞
)
(3.4)
The wing-fixed forces are simply scaled by the instantaneous velocity increase, since
the force production is proportional to the total velocity but the force normalization
is based only on the freestream velocity. The flow-fixed forces need to be scaled and
rotated, since C˜L is normal to the total velocity but CL is normal to the freestream
velocity.
CL(t) =
(
C˜L(α+θ(t)) cos(θ(t)) + C˜D(α+θ(t)) sin(θ(t))
)(
1+
V (t)2
U2∞
)
(3.5)
CD(t) =
(
C˜D(α+θ(t)) cos(θ(t))− C˜L(α+θ(t)) sin(θ(t))
)(
1+
V (t)2
U2∞
)
(3.6)
The static force measurements were obtained from a gust-off angle-of-attack sweep
at Re = 20,000. The results of this model can be compared to measured forces for
57
0 10 20 30 40
 
0
0.5
1
1.5
2
2.5
Measured
Prediction
0 10 20 30 40
0
0.5
1
1.5
2
2.5
0 10 20 30 40
0
0.5
1
1.5
2
2.5
0 10 20 30 40
0
0.5
1
1.5
2
2.5
Figure 3.6: Quasi-steady gust force prediction based on static force data.
α = 45° and G = 0.84 in figure 3.6. The steady and peak forces are closely matched,
but the gradual gust recovery is completely missed — as was anticipated due to the
inherent limits of the quasi-steady model.
3.2.1 Adaptations of indicial theory
The transverse gust encounters studied here were intentionally significantly
unsteady events. The lingering effects of the gust encounter over the roughly 30
chord-lengths after the wing has exited the gust indicates that the force production
is (as expected) impacted by the flow history rather than just the instantaneous
conditions. Because of this, unsteady aerodynamic models may better capture the
gust encounter force production.
58
Indicial theory allows the impulse response of a system to be extended to
model the response to arbitrary forcing by approximating that forcing as many small
impulse changes. Ku¨ssner’s function is the impulse response of a two-dimensional
flat plate wing to an encounter with a small-perturbation sharp-edged gust. By using
Duhamel’s integral, the convolution of the impulse response with the slope of the
arbitrary forcing, Ku¨ssner’s function can be extended to spatially distributed gusts.
The typical application of Duhamel’s integral and Ku¨ssner’s function is based
on thin-airfoil theory. The validity of this method for transient force production
during high-amplitude gust encounters will be discussed below, but it is at least clear
that thin-airfoil theory is not applicable during the steady-state conditions preceding
the gust encounter. To proceed with this method, the steady-state force production
must be made to match the separated flow conditions by substituting empirical
results for the initial condition, CN, static(t
∗ = 0). All three adaptations tested in the
current work include this change. Note that in the small-perturbation scenarios for
which thin-airfoil theory was derived, lift and plate-normal force are approximately
equivalent, and the results are typically presented as lift. In the current work, normal
force was chosen for these large-perturbation adaptations so that the thin-airfoil
theory results approach the more reasonable solution CN(90°) = 2pi sin(90°) = 2pi
rather than CL(90°) = 2pi when the measured CL(90°) = 0.
The variations in the three present adaptations are contained in the unsteady
term, in which the arbitrary forcing in Duhamel’s integral is expressed in a manner
that captures the aerodynamic response of the system, CN, response. This general form
59
of the indicial method using Ku¨ssner’s function adapted for large gusts is therefore:
CN(s) = CN, static(0) +
∫ s
0
[
d
dσ
CN, response(σ)
]
ψ(s−σ) dσ (3.7)
In this equation, CN(s) is the resulting unsteady normal force coefficient. As stated
above, CN, static(0) is the empirical normal force coefficient measured at the same
wing pitch angle, which is used here as the initial condition so that the models agree
with the measured force histories before and after the gust encounter. ψ(s− σ) is
Ku¨ssner’s function, which was derived for small-perturbation sharp-edged gusts, but
is the closest available indicial response for the large-amplitude gusts of interest
in the current work. CN, response(σ) is a placeholder which will be replaced by some
function of the quasi-steady aerodynamic response of the wing. The independent
variable here is s = 2tU∞
c
, the semi-convective time. The final variable, σ, is a bound
variable for integration over all values of s in the flow history.
In the usual small-angle application of the indicial method for a gust response
(see equation 1.12), the forcing, CN, response, is expressed as in linearized thin airfoil
theory, CN ≈ CL = 2pi sin(θ) where θ = α+ tan
(
V(t)
U∞
)−1
. This can be linearized by
sin(θ) ≈ θ and tan
(
V(t)
U∞
)−1
≈ V(t)
U∞ . In the current work, a correction is incorporated
for the finite aspect ratio of the test article: dCL
dα
= 2pi
1+ 2
A
= 4pi
3
[7]. This unsteady
force production model will be referred to here as the linearized unsteady thin-airfoil
model, or in figure legends as “linearized.”
CN(s) = CN, static(0) +
2pi
U∞
(
1+ 2
A
) ∫ s
0
[
d
dσ
V (σ)
]
ψ(s−σ) dσ (3.8)
60
Only one term of linearized thin-airfoil theory is not constant, so much of this model,
2pi
U∞(1+ 2
A
)
, can be moved outside of the integration. The unsteady aerodynamics here
are expressed directly as a function of the gust velocity profile, V (σ).
The use of linearized thin-airfoil theory is typical because the flat-wake
assumption necessary for Ku¨ssner’s function means that small-angle approximations
are valid. However, the simplification of the general form which converted equation
3.7 into equation 3.8 does not need to be taken so far. By using thin airfoil theory
without the linearizing small-angle approximations, the model can be applied to
higher incidence angles (caused by higher gust ratios) without the additional error
introduced by linearizing. Those same high incidence cases violate the flat-wake
and attached flow assumptions built into Ku¨ssner’s function, but this is true of
each model described here. If the quasi-steady aerodynamic response of the system
is modeled using thin-airfoil theory without the small-angle approximations, the
resulting equation is:
CN(s) = CN, static(0) +
2pi
1+ 2
A
∫ s
0
[
d
dσ
sin
(
α + tan
(
V (σ)
U∞
)−1)]
ψ(s−σ) dσ (3.9)
As with equation 3.8, the constant parts of the lift-curve slope, 2pi
1+ 2
A
, can be removed
from the integration. In equation 3.9, however, much more remains in the unsteady
term. The forcing function of Duhamel’s integral in this case is not the gust
velocity profile as before; the forcing is the sine of the effective flow incidence
angle, sin
(
α + tan
(
V(σ)
U∞
)−1)
. While both the linear and non-linear versions of this
model incorporate the initial conditions into CN, static(0), only the non-linear model
61
incorporates those conditions into the unsteady portion by depending on α. This
unsteady force production model will be referred to here as the non-linear unsteady
thin-airfoil model, or in figure legends as “non-linear.”
The theoretical aerodynamics of thin-airfoil theory are not a required component
of this indicial convolution method. The static force curve, CN ≈ CL = 2pi sin(θ),
can be replaced entirely by the empirical quasi-steady response. This adaptation
incorporates the features of the static flow into the unsteady model, such as flow
separation and the static force curve, CN(θ). However, those features do not remain
unchanged between the static and unsteady cases, so this model is still only an
approximation of the unsteady behavior.
CN(s) = CN, static(0) +
∫ s
0
[
d
dσ
CN, static(σ)
(
1 +
(
V (σ)
U∞
)2)]
ψ(s−σ) dσ (3.10)
In equation 3.10, the quasi-steady aerodynamic response is represented by an
empirical look-up table of force coefficients, CN, static(σ), and a relative variation
in total dynamic pressure,
(
1 +
(
V(σ)
U∞
)2)
. This third unsteady force production
model will be referred to here as the semi-empirical unsteady model, or in figure
legends as “semi-empirical.”
The force history predictions made by these models can be compared to
measurements in figure 3.7. Each plot shows the force history for one gust encounter,
which are grouped into columns by wing pitch angle, and into rows by gust ratio.
Four models are compared to each measured force history: first the quasi-steady
semi-empirical model, then the three adaptations of the indicial method. All of the
62
models predict force histories which are vaguely correct in magnitude and shape,
although the quasi-steady model does not capture the gust recovery behavior, and
each model is up to 50% off of the peak force magnitude in some of the cases.
The use of small-angle approximations causes the linearized and non-linear
unsteady thin-airfoil models to predict similar results for low angles and gust ratios,
and different results for higher angles and gust ratios. The non-linear unsteady
thin-airfoil model most closely matches the results except for the highest gust
ratio. In that case, the effective incidence angle passed plate-normal and peaked
at α + θ = 104°. The behavior of the non-linear unsteady model did not remotely
match the measured results in that case. However, the unsteady semi-empirical
model matched the measured results for the high gust-ratio case quite well. The
quasi-steady semi-empirical model predicts the peak force production at the correct
time in each case, while all of the indicial methods lag behind the measured behavior.
The local minimum in force production as the wing exits the gust poses an interesting
challenge for future work as it is not captured by any of the current models. In
fact, each of the present models is inherently incapable of predicting that particular
feature, since the unsteady models are all based on the indicial response given by
Ku¨ssner’s function which does not predict overshoot under any circumstances.
In the various gust encounters presented in figure 3.7, different models match
most closely to the measured forces depending on the parameters. None of the
models accurately capture the peak force for all of the six cases. Also, none of the
models fully capture the gust recovery process, although the indicial methods mirror
the overall behavior over a shorter time span.
63
-5 0 5 10 15 20
0
1
2
3
4
5
-5 0 5 10 15 20
0
1
2
3
4
5
-5 0 5 10 15 20
0
1
2
3
4
5
-5 0 5 10 15 20
0
1
2
3
4
5
-5 0 5 10 15 20
0
1
2
3
4
5
-5 0 5 10 15 20
0
1
2
3
4
5
Figure 3.7: Comparison of measured to predicted normal force coefficients
for six measurements and four models.
64
3.3 Variations in peak force production
The variations in force histories and predictions across the measured parameter
space can be analyzed in a more quantitative manner if the data is reduced to a
set of characteristic scalar quantities. Each of the 140 measured force histories was
characterized by the peak values and steady values of lift and drag coefficients — or
equivalently and interdependently: plate-normal and plate-tangent force coefficients.
The steady forces in each case were taken as the mean force produced over the final
25 chord-lengths of constant towing velocity, 36 < t∗ < 61. The peak force in each
case was taken as the maximum force measured during the gust encounter. To avoid
exaggeration of maximum forces due to measurement noise, the force histories were
smoothed using weighted quadratic local regression (“loess” in Matlab) with a span
equal to half of the gust width before finding the peak value. The characteristic
values of force coefficients are demonstrated in figure 3.8.
The steady forces after gust recovery are plotted in figure 3.9 as black circles.
As expected, all of these measurements agree with the static angle-of-attack sweep
data represented by the black line. These points have been plotted with some
transparency so that the varying saturation of color can qualitatively indicate the
overlap of repeated measurements.
The peak gust forces can be compared similarly by accounting for the increased
total velocity and relative angle due to the gust. For a pitch angle, α, freestream
velocity, U∞, and gust velocity, Vg, the peak effective angle of attack and total
65
0 10 20 30 40 50 60
 
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50 60
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50 60
 
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50 60
 
0
0.5
1
1.5
2
2.5
Figure 3.8: Each force history was reduced to a set of characteristic steady
and peak forces.
velocity are:
αpeak = α + tan
(
Vg
U∞
)−1
(3.11)
Vtotal =
√
U2∞ + V 2g (3.12)
The peak forces are plotted at αpeak on the x-axis and normalized by Vtotal. The lift
and drag coefficients are decomposed relative to αpeak. This is the reverse of the
process used for the empirical quasi-steady model described in section 3.2. These
re-normalized peak forces are plotted as diamonds colored by the associated gust
ratio. Despite the rapid change of conditions around the wing during the gust
encounter, the measured peak force coefficients were found to be close to the static
measurements for a wide range of conditions.
66
0 20 40 60 80 100
 
-0.5
0
0.5
1
1.5
2
0 20 40 60 80 100
0
0.5
1
1.5
2
0 20 40 60 80 100
 
-0.5
0
0.5
1
1.5
2
0 20 40 60 80 100
0
0.5
1
1.5
2
Figure 3.9: Quasi-steady analysis of peak unsteady gust forces.
It is apparent based on deviations between the steady and unsteady data
(particularly in normal-force coefficient) that some unsteady effect is present that is
not accounted for by the empirical quasi-steady analysis. The influence of unsteady
effects is further observed by recalling the trend in peak forces in figure 3.4, which
in disagreement with static data showed decreasing peak normal force coefficients
with increasing angles of attack between 20° and 45°. This is visible in the CN axis
of figure 3.9 by noting that the right-most points of each color descend as the angle
increases, but the black line that shows static force coefficient increases (if unevenly)
for all angles up to α = 90°.
The difference between the geometric pitch angle of the wing and the peak
effective angle-of-attack during the gust is (non-linearly) proportional to the gust
67
0 20 40 60 80 100
 
-0.5
0
0.5
1
1.5
2
0 20 40 60 80 100
0
0.5
1
1.5
2
0 20 40 60 80 100
 
-0.5
0
0.5
1
1.5
2
0 20 40 60 80 100
0
0.5
1
1.5
2
Figure 3.10: Convergence of peak forces for gust encounters with different
initial conditions (a subset of the data from figure 3.9).
ratio. Cases with identical pitch angles, represented by overlapping black circles in
figure 3.9, but different gust ratios result in different peak relative angles, represented
by colored diamonds spread out across the x -axis. Several cases are particularly
interesting; these cases have different pitch angles and gust ratios that cause the peak
relative angles to coincide. Specifically, cases at α = 45° with a gust ratio of 0.42
and α = 7° with a gust ratio of 1.68 both result in a peak effective angle-of-attack
within 1° of αpeak = 67°. It can be seen in figure 3.10 that the re-normalized peak
lift and drag in these cases agree, suggesting that these forces depend on the peak
quasi-steady conditions and not the pitch angle or gust ratio independently.
68
3.3.1 Unsteady analysis of peak force
The influence of unsteady aerodynamics on the peak forces can be investigated
using the indicial models given by equations 3.8, 3.9, and 3.10. The peak normal
forces predicted by the adaptations of Ku¨ssner’s function were characterized and
compared to the measurements using the same procedure shown in figure 3.8. Figure
3.11 shows these peak predicted values, although they are plotted differently than
the measurements. The peak value for each indicial model is a function of only the
initial pitch angle, the gust ratio, and the gust width. In the current work only one
gust width was tested, so the remaining two-dimensional variation in peak force
predictions was plotted as one line for each gust ratio (colored to match the measured
results for each tested gust ratio).
The linearized unsteady thin-airfoil model (dotted lines) matches the magnitude
of the forces well for each gust ratio, but does not capture the proper trends with
respect to pitch angle. This is particularly visible for the highest gust-ratio cases
(green), for which the measured peak forces decrease with increasing pitch angle but
the predicted peak forces increase with increasing pitch angle.
The non-linear unsteady thin-airfoil model (solid lines) under-predicts the force
magnitudes as the gust ratio increases, but matches the trend with respect to pitch
angle very closely. If the results for the middle and higher gust ratio cases (red and
green) were transposed upward by the appropriate amount, this model could fit very
closely to the measured peak force for every case in the current work. This seems to
indicate that the effect of flow incidence angle on peak force production is recreated
69
0 20 40 60 80 100
-0.5
0
0.5
1
1.5
2
Figure 3.11: Adaptations of Kussner’s function compared to measured peak
normal force coefficients.
accurately by the non-linear unsteady thin-airfoil model, but that the effect of local
dynamic pressure — which changes with gust ratio — is mis-characterized.
The unsteady semi-empirical model (dashed lines) does not match the
magnitudes or the trends of the measured peak forces. This model seems to under-
predict the increase in lift due to both the flow incidence angle and the local dynamic
pressure. The under-prediction with respect to flow incidence angle in particular
supports the interpretation that the unsteady force production is better modeled by
the attached flow of theoretical aerodynamics rather than the separated flow of the
static data.
70
3.4 Chapter summary
In this chapter, measured forces were presented and analyzed for sine-squared
transverse gust encounters of width w = 3.2c, gust ratios G = [0.42, 0.84, 1.68], and
wing pitch angles between −4° ≤ α ≤ 45°. These gust encounters created large
transient forces which were observed to coincide with vortex formation at the leading
edge during augmented lift and at the trailing edge during reduced lift.
The magnitudes of the peak forces were compared to predictions made using a
quasi-steady empirical model and large-perturbation adaptations of indicial theory
using Ku¨ssner’s function. The peak predicted forces did not perfectly agree with
measurements for any of the available models, but the general magnitude was matched
by the quasi-steady empirical model and the linearized unsteady thin-airfoil model.
The shape of the force trend with respect to pitch angle within each gust ratio
was matched very closely by the predictions of indicial theory using the non-linear
unsteady thin-airfoil model.
71
Chapter 4: Plunging wing force production
This chapter presents the measured force production by a thin, flat, rectangular
wing performing single plunging maneuvers. These results serve to test aerodynamic
models which do not rely on the harmonic nature of those in previous studies. The
maneuvers vary in horizontal and vertical extent, duration, and wing pitch angle.
The details of the tested parameter matrix are presented in section 2.5.
The force production during these plunge maneuvers was measured and
qualitative trends with respect to each of the independent parameters are discussed.
A data reduction procedure was then used to identify quantitative trends in force
production across multiple axes of the parameter space. Additionally, force production
was compared between the plunge maneuvers and the transverse gust encounters.
4.1 Plunging wing force history features
Figure 4.1 shows the force coefficient histories for four differently-shaped plunge
maneuvers. The forces produced in each case are presented as lift and drag coefficient
normalized first by the freestream dynamic pressure and then by the mean plunge
72
0 1 2 3 4 5 6
-10
0
10
20
30
40
0 1 2 3 4 5 6
-10
0
10
20
30
40
0 1 2 3 4 5 6
-5
0
5
10
0 1 2 3 4 5 6
-5
0
5
10
Figure 4.1: Example measurements of lift coefficient during the plunge
maneuvers. In each case, α = 0°.
kinetic energy (see section 2.6.3):
CL =
L
1
2
U2∞ρS
(4.1)
Lnorm =
L
1
2
(
U2∞ +
3
8
V 2p
)
ρS
=
CL
1 + 3
2
(
h
w
)2 (4.2)
There are several striking features of these force histories. First, the magnitude
of the lift coefficients is unreasonably high; it is important to keep in mind that this is
an artifact of the freestream-dynamic-pressure method of normalization rather than
a true indication of lift efficiency. The mean-kinetic-energy normalization attempts
73
to correct this, although the magnitudes of the coefficients are still high. Second, the
force histories vary substantially in shape, even though the wing kinematics are all
variations of the same shape. This supports the expectation that multiple physical
sources of force production are involved. Third, as was also observed for this flat
plate wing in the gust, force production is dominated by plate-normal pressure forces.
Plate-tangent skin friction forces are negligible. The wing pitch angle in each case
shown here is 0°, so the lift is equivalent to the plate-normal force and the drag is
equivalent to the tangent force. Those components will be discussed separately for
cases with pitch angles above zero. Finally — and most surprisingly — there is very
little unsteady force production after the plunge maneuver has ended. Unlike the
extremely long recovery process from the gust encounters, the wing appears to return
to steady force production within one chord-length traveled. This suggests that
quasi-steady analysis may be a reasonable approach despite the large unsteadiness
of the plunge maneuver itself.
The plunge maneuver kinematics associated with the cases displayed in figure
4.1 vary in both width and height, and no clear trend in force production with respect
to kinematics is visible from this data. Of these four cases, the largest peak lift
coefficient is observed during different kinematics depending on the normalization
method: the highest CL in figure 4.1 is 33 in the w = 2c, h = 3c case, but the highest
Lnorm is 9.4 in the w = 1c, h = 1c case. This is because the mean-kinetic-energy
normalization incorporates the plunge velocity component, which suitably reduces
the normalized forces for cases which have high a plunge velocity ratio. The lowest
lift coefficient is observed for the same kinematics in each normalization, w = 6c and
74
h = 3c.
4.2 Trends with respect to plunge kinematics
To understand how the variations in force production are related to variations
in each of the plunge maneuver parameters, comparisons will be made between force
histories measured for cases which vary in only one of these parameters at a time. It
is also useful to identify how these maneuver parameters affect the primary physical
quantities associated with force production: acceleration and dynamic pressure. The
plunge maneuver kinematic used in all cases of the current study was the sine-squared
gust profile:
V(t) = Vp sin
(
pit
T
)2
for 0 ≤ t ≤ T (4.3)
This can be rewritten based on the distances traveled during the maneuver, h = 1
2
VpT
and w = U∞T , as:
V(t) =
2hU∞
w
sin
(
pitU∞
w
)2
(4.4)
The plunge acceleration during the maneuver can then be written as:
V˙(t) =
4hU2∞
w2
sin
(
pitU∞
w
)
cos
(
pitU∞
w
)
(4.5)
The horizontal velocity, U∞, is constant, so it does not contribute to the acceleration.
The freestream velocity does contribute to the peak dynamic pressure, though:
1
2
ρU2max =
1
2
ρ
[
U2∞ +
(
2hU∞
w
)2]
=
1
2
ρU2∞
[
1 +
(
2h
w
)2]
(4.6)
75
0 1 2 3 4 5 6
-10
0
10
20
30
0 1 2 3 4 5 6
-10
0
10
20
30
0 1 2 3 4 5 6
-5
0
5
10
0 1 2 3 4 5 6
-5
0
5
10
Figure 4.2: Measurements of forces during plunge maneuvers which vary
only in freestream Reynolds number.
The proportional variations of acceleration and dynamic pressure with respect to
each of the maneuver parameters, h, w, and U∞, will help to decipher the observed
effects that each of these parameters has on the measured force production.
4.2.1 Effects of varying Reynolds number
Figure 4.2 shows lift coefficient histories for cases which vary only in Reynolds
number, which determines the freestream velocity and thus also the maneuver
duration. Both the peak acceleration experienced by the wing and the peak dynamic
pressure increase proportionally with the square of the freestream velocity. This
proportionality is precisely why lift coefficient is typically normalized by freestream
76
dynamic pressure, and as seen here the normalized forces are not changed by the
varied Reynolds number.
Figure 4.2 only shows the comparison for the w = 1c, h = 1c case, but each
other set of cases which only varied by Reynolds number also resulted in matching
force coefficients. This is true for both CL and Lnorm, since the difference between
these normalizations does not vary with Reynolds number. These results agree
with previous conclusions that force production by thin flat plates is insensitive to
Reynolds number [27]. This result is useful primarily in reducing the parameter
space to the more manageable size of three independent parameters, and for allowing
comparisons of maneuvers with different Reynolds numbers.
4.2.2 Effects of varying plunge height
The peak acceleration of the wing increases linearly proportionally with
the maneuver height. The variation in peak dynamic pressure with respect to
maneuver height is more complex since it depends on both the horizontal and vertical
velocities. This is displayed in figure 4.3. The peak dynamic pressure normalized
by the freestream dynamic pressure is shown in blue on a logarithmic plot. The
contribution to peak dynamic pressure by the vertical velocity, shown in red, increases
proportionally with the square of the maneuver height. The contribution by the
horizontal velocity, shown in green, does not increase at all. This causes the variation
in peak dynamic pressure with respect to plunge height, d
dh
(qmax), to approach
quadratic for high velocity ratios, but to fall below the quadratic variation for lower
77
0.1 1 10
1
10
Figure 4.3: Variations in dynamic pressure contributions by U∞ and Vp with
respect to plunge velocity ratio.
velocity ratios.
Figure 4.4 shows force coefficient histories for cases which vary in plunge
maneuver height. These cases also vary in freestream Reynolds number (they
were defined so that the peak velocity matched instead), but it was shown in the
previous section that variations in Reynolds number did not change the normalized
force histories. The lift coefficient axis (top left) of figure 4.4 clearly shows that
increasing the plunge height increases the normalized force production. The increased
plunge height (when the width is fixed) corresponds to an increase in the relative
contribution of the vertical velocity component compared to the horizontal velocity.
The normalization of forces by freestream dynamic pressure does not account for this,
so the normalized force is bound to increase. Interestingly, the mean-kinetic-energy
lift normalization collapses four of the six curves, which means that the variation
in force production with respect to plunge depth is proportional to the variation in
78
0 1 2 3 4 5 6
-10
0
10
20
30
40
50
0 1 2 3 4 5 6
-10
0
10
20
30
40
50
0 1 2 3 4 5 6
-5
0
5
10
0 1 2 3 4 5 6
-5
0
5
10
Figure 4.4: Measurements of forces during plunge maneuvers which vary in
height but not width.
mean kinetic energy for these cases. The two lowest-amplitude cases fall below the
trend. This is related to the discussion of variations in peak dynamic pressure with
respect to plunge height, since the freestream dynamic pressure and the mean kinetic
energy are related by a similar nearly-quadratic term (see section 2.6.3): 1 + 3
2
(
h
w
)2
.
For the four higher velocity ratio cases here, Vp
U∞ =
2h
w
>= 2, and so the variations in
both the peak dynamic pressure and the mean kinetic energy begin to converge to
the same quadratic trend.
79
4.2.3 Effects of varying plunge width
The peak acceleration of the wing increases proportionally with the inverse
square of the maneuver width: ∝ 1
w2
. A wing in a wider plunge maneuver experiences
less acceleration — as long as the height is fixed. The peak dynamic pressure has a
similar (but inverse) behavior with respect to plunge width to that which it had with
respect to plunge height. The relative contribution of peak dynamic pressure by the
horizontal velocity increases proportionally with the square of the maneuver width,
but the contribution by the vertical velocity does not increase at all. In terms of
equation 4.6, the peak dynamic pressure approaches the freestream dynamic pressure
for very wide maneuvers, and is higher than the freestream dynamic pressure for
narrower maneuvers.
Figure 4.5 shows force coefficient histories for cases which vary in plunge
maneuver width (and Reynolds number, but as previously discussed, this did not
affect the force coefficients). This figure is a good demonstration of the weakness of
the typical lift coefficient normalization by freestream dynamic pressure. The first
case plotted here peaks at CL ≈ 120 and so had to be truncated so that the other
five cases would be visually discernible. This amplitude difference is primarily caused
by the relative magnitudes of the vertical and horizontal velocity components, and so
that difference is minimized by the mean-kinetic-energy lift normalization. In both
plots, it is clear that the peak normalized lift decreases with increasing maneuver
width, as do both the peak acceleration and dynamic pressures experienced by the
wing during the maneuver. As is particularly apparent in figure 4.5, it is difficult to
80
0 1 2 3 4 5 6
-10
0
10
20
30
40
50
0 1 2 3 4 5 6
-10
0
10
20
30
40
50
0 1 2 3 4 5 6
-5
0
5
10
0 1 2 3 4 5 6
-5
0
5
10
Figure 4.5: Measurements of forces during plunge maneuvers which vary in
width but not height.
say quantitatively how the magnitudes of these force histories vary because they also
vary significantly in overall shape. This is addressed below in section 4.3.1 in order
to better quantify the trends in force production with respect to plunge maneuver
height and width.
4.2.4 Effects of varying wing pitch angle
The geometric pitch angle of the wing does not appear in the kinematic
equations for either acceleration (equation 4.5) or dynamic pressure (equation 4.6),
which were suggested to be the key contributors to force production. However, the
variation in force production with respect to acceleration and dynamic pressure are
81
0 1 2 3 4 5 6
-5
0
5
10
15
20
0 1 2 3 4 5 6
-5
0
5
10
15
20
0 1 2 3 4 5 6
-5
0
5
10
15
20
0 1 2 3 4 5 6
-5
0
5
10
15
20
Figure 4.6: Measurements of forces during plunge maneuvers which vary in
wing pitch angle but not width or height.
dependent on other quantities as well. Some of these, such as fluid density, ρ, and
wing planform area, S, were not varied in the current work. The wing pitch angle, α,
was varied here in order to inspect its effects on unsteady force production. During
the plunge maneuver, the relative incidence angle of the fluid is the sum of the wing
pitch angle and the angle of the velocity vector of the wing, α + tan
(
V(t)
U∞
)−1
.
Figure 4.6 shows force coefficient histories for cases which vary in wing pitch
angle, but are otherwise the same maneuver kinematics. These α 6= 0° cases separate
the lift from the normal force, and so further verify that the force production
is primarily plate-normal. Somewhat surprisingly, each of the four plotted cases
produced nearly the same normal force histories. The lift and drag projections of
normal force change with pitch angle, but the plate-normal forces are not significantly
82
affected by the wing pitch angle for the tested range, 0° ≤ α ≤ 45°.
In these example cases, the plunge maneuver increases the flow incidence angle
by tan
(
2h
w
)−1
= 63° at the peak. It should be expected then that the case which
began at α = 0° and increased to α + θ = 63° would produce different transient
forces than that which began at α = 45° and increased to α + θ = 108°. The change
in quasi-steady force production, dCN
dα
, certainly varies between these two ranges (as
shown in figure 2.4). Instead, it was observed that these cases — and those for wing
pitch angles in between — resulted in similar force histories. This suggests that the
initial pitch angle has very little impact on the overall transient force slope, dCN
dθ
,
although the mechanism behind that behavior is unknown.
The initial pitch angle does impact the steady state normal force before and
after the plunge, but this effect is negligible in figure 4.6 due to the relative magnitude
of the steady and unsteady force contributions. Even in these moderate plunge cases
(w = 3c, h = 3c) the peak force coefficients during the maneuver reach CN ≈ 15,
which is much greater than any normal force coefficient measured outside of the
maneuver. The steady-state normal force coefficient at the highest angle tested here,
α = 45°, was CN = 1.2, only 8% of the peak observed value in figure 4.6.
It can be seen from equation 1.6 that the inertial component of unsteady forces
should also vary with respect to wing pitch angle. That equation can be simplified
for the current maneuver to show inertial normal force due to plunge acceleration:
CN, inertial = − pic
2U2∞
V˙ cos(α) (4.7)
83
This potential flow model indicates that the normal force production is proportional
to the chord-normal projection of the acceleration, V˙ cos(α), which depends on wing
pitch angle. This can be reconciled with the observed force histories by recognizing
that both the relative change in cos(α) and the magnitude of inertial forces are
small compared to the peak observed forces. The chord-normal component of plunge
acceleration is 100% at α = 0°, 98% at α = 10°, 94% at α = 20°, and 71% at α = 45°.
Therefore, in figure 4.6, only the case at α = 45° experiences a significantly different
chord-normal acceleration from the others, although even in this case there is no clear
difference in observed force production. Equation 4.7 predicts that the peak added-
mass force for w = 3c, h = 3c, α = 0° is CN, inertial = 3.3, which is only 22% of the
observed peak force. Assuming for the moment that this model accurately represents
the inertial force production of the plunging wing, the non-inertial forces are far
greater than the inertial forces in figure 4.6. This can be qualitatively corroborated
by observing the high force production at t∗ = 1.5, since at that moment the plunge
acceleration is V˙ = 0 and so no inertial force is expected. Since the magnitudes of
the steady and inertial forces are small compared to the total unsteady force, their
known variations with respect to wing pitch angle does not necessarily contradict
the observed lack of variation in total force.
Figure 4.6 only shows cases for which w = 3c, h = 3c, but these results held
for other plunge kinematics as long as the steady-state forces were small compared
to the unsteady forces. The normal force histories of each tested maneuver size (for
which α was varied) are shown in figure 4.7.
The variations in steady-state force production are only easily apparent in the
84
Table 4.1: Peak steady normal force coefficient (1.2) as a percent of peak
observed force coefficient for plunge maneuvers in figure 4.7.
CN, steady
CN, peak
w = 1c 3c 6c
h = 1c 5.7% 30% 60%
3c 0.92% 8.0% 22%
6c 0.32% 2.9% 9.2%
Table 4.2: Added-mass normal force coefficient as a percent of peak observed
force coefficient for plunge maneuvers in figure 4.7.
CN, inertial
CN, peak
w = 1c 3c 6c
h = 1c 47% 27% 14%
3c 23% 22% 15%
6c 16% 16% 13%
three lowest velocity ratio cases (the top-right and adjacent plots). This observation
is in agreement with the relative magnitudes of the steady and unsteady forces for
these cases, shown in table 4.1. Only in those three cases were the unsteady forces
low enough that the differences in steady-state force production due to wing pitch
angles were not negligible. In all of the other five cases, the normal force histories
were not significantly affected by changes in wing pitch angle.
Similarly, table 4.2 shows the relative magnitudes of the added-mass and total
unsteady forces for these cases. Most of these cases are only slightly influenced by
inertial forces. The top-left axis of figure 4.7 shows forces for the case of w = 1c,
h = 1c, for which the potential flow model predicts that the added-mass force —
which, according the model, ought to vary with respect to wing pitch angle — is
the highest of these cases at 47% of the peak observed force. Despite this greater
inertial contribution than in the w = 3c, h = 3c case from figure 4.6, no obvious
force variation with respect to wing pitch angle is visible.
85
0 0.5 1 1.5
-20
0
20
0 0.5 1 1.5
-50
0
50
100
150
0 0.5 1 1.5
-200
0
200
400
0 2 4
0
2
4
0 2 4
-5
0
5
10
15
0 2 4
0
20
40
0 5 10
0
1
2
0 5 10
0
2
4
6
0 5 10
0
5
10
15
Figure 4.7: Measurements of normal force coefficient during several plunge
maneuvers. Each axis shows a group which varies in wing pitch angle but
not width or height. Note that the axes of each are scaled differently.
86
4.3 Force history characterization
As was done for the gust encounter data, plunge maneuver force production
can be more usefully quantitatively analyzed if the force histories are reduced to a
few characteristic scalars. This could be accomplished by extracting peak unsteady
force production, but as is particularly evident in the Lnorm axis of figure 4.5, the
force production during the plunge maneuver has two constituent shapes which
blend between the cases. In order to capture the behavior of these two separate
components, the unsteady force production will be characterized by two scalars
instead of one.
4.3.1 Two-term decomposition
As was noted above, force histories for these sine-squared plunge maneuvers
seem to contain two distinct constituent shapes. Force histories for plunge maneuvers
which are much taller than they are wide, h
w
 1, are mainly the shape of a sine
wave. The shallower plunge cases, h
w
 1, result in force histories which are mainly
a one-minus-cosine shape. Each force history can be modeled as the sum of these
two shapes scaled by some coefficients, C1 and C2, and some residual, R. While the
wing is plunging (0 ≤ t∗ ≤ w):
FN(t
∗) ≈ C1
[
1− cos
(
2pit∗
w
)]
+ C2 sin
(
2pit∗
w
)
+ R (4.8)
87
The functions can be related back to the two primary sources of force production:
acceleration and dynamic pressure. For the sine-squared kinematics of the current
study, acceleration and dynamic pressure, given by equations 4.4 and 4.5, are these
same two general shapes (the sine-squared plunge maneuver tested here is equivalent
to one-minus-cosine by 1− cos(2θ) = 2 sin(θ)2). The resulting simplified model for
normal force production, FN , is then a scaled contribution from dynamic pressure
plus another scaled contribution from acceleration. Any possible higher order effects
are combined in the residual. This model can be fit to each measured force history
by finding values for the coefficients which minimize the residual using least-squares
regression. This general model is expressed in equation 4.9, and the shape of each
term is isolated in equation 4.10. The latter equation was the one used for the
least-squares curve fit.
FN(t
∗) = −C1V (t∗)2 − C2V˙ (t∗) + R (4.9)
FN(t
∗) = C1 sin
(
pit∗
w
)4
+ C2 sin
(
pit∗
w
)
cos
(
pit∗
w
)
+ R (4.10)
This equation was fit to each of the measured force histories to characterize them by
the two coefficients, C1 and C2, and to quantify the minimum residual, R. To avoid
confusion, it is worth stating that the regression process was applied to the non-
normalized force data (in Newtons), although each normalization option is a linear
operation and so would not affect the outcome. The coefficients were normalized by
either method after the force history characterization.
88
Figures 4.8 and 4.9 show examples of this decomposition model fit to measured
force histories. Although the two example cases are very different, the model fits
closely to each set of measured forces. The plunge maneuver which produced the
forces in figure 4.8 was 1c in both width and height — an aggressive plunge maneuver.
The coefficients normalized by mean kinetic energy were C1 = 4.9 and C2 = 6.2. In
this case, the acceleration was responsible for slightly more force production than
was the variation in dynamic pressure. The root-mean-square (rms) of the residual
was 0.9, which is acceptably small compared to the two characteristic terms. The
other example, shown in figure 4.9, was a much more gentle plunge maneuver of
width 6c and height 3c. The coefficients normalized by mean kinetic energy were
C1 = 4.0 and C2 = 0.1. In this case, the acceleration was responsible for far less
force production amplitude than the dynamic pressure. The rms of the residual was
0.4, which is again acceptably small even though C2 is also quite small in this case.
This process was repeated for each of the 108 measured plunge kinematics for
which α = 0°. In that way, the force production variations with respect to both
maneuver width and height were quantified in terms of C1, C2, and rms(R). Figure
4.10 shows those results — the same data is plotted in each axis, but it is displayed
differently in each. The top row shows CL and the bottom row shows Lnorm. The left
column is linearly scaled and the right column is logarithmically scaled (x -axis only).
The x -axis for this figure is plunge maneuver aspect ratio, h
w
, because it appeared as
an important term in the equations for both the peak acceleration and peak dynamic
pressure during each maneuver. It is visible (and unsurprising) that this parameter
only incompletely captures the variation in plunging wing force production across
89
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
-6
-4
-2
0
2
4
6
8
10
12
Measured lift
Least-squares fit
Velocity term, C1
Acceleration term, C2
Residual
Figure 4.8: Best fit of the two-term lift decomposition for a high velocity
ratio plunge maneuver: w = 1c, h = 1c.
the parameter space.
The linear axis of CL (top left) shows that — at least for high aspect ratio,
aggressive plunge maneuvers — both contributions to force production increase
vaguely quadratically. This is corroborated by both the linear and logarithmic axes
of Lnorm, which shows that the mean-kinetic-energy-normalized force production is
roughly constant for plunge maneuver aspect ratios above one. The roughly constant
values of Lnorm can be justified by the same proportionality to dynamic pressure
that caused most of the data in figure 4.4 to converge for the mean-kinetic-energy
normalization. The “banding” of smaller groups of data points within the overall
trend — particularly visible in C1 on the logarithmic axis of Lnorm (bottom right) —
suggests that other parameter variations are also responsible for variations in plunge
90
0 1 2 3 4 5 6
-1
0
1
2
3
4
5
Measured lift
Least-squares fit
Velocity term, C1
Acceleration term, C2
Residual
Figure 4.9: Best fit of the two-term lift decomposition for a low velocity
ratio plunge maneuver: w = 6c, h = 3c.
force production. Additionally, it is clear that there are outliers in the C2 amplitudes
using Lnorm (at
h
w
= 1, there are C2 observations ranging from 2 to 13) which also
indicates variations with respect to other plunge maneuver parameters. To identify
more complete characterizations of these variations, the characteristic amplitudes
can be compared to those predicted by various analytical models.
4.3.2 Comparison to analytical models
The small-perturbation solution for unsteady force production by a moving wing
by von Ka´rma´n and Sears [18] divided total force production into three components:
quasi-steady, inertial, and wake contributions (see section 1.2.4). The division of
91
0 2 4 6 8 10
0
200
400
600
800
1000
10-1 100 101
0
200
400
600
800
1000 Velocity term, C1
Acceleration term, C2
rms Residual
0 2 4 6 8 10
0
2
4
6
8
10
12
14
16
10-1 100 101
0
2
4
6
8
10
12
14
16
Figure 4.10: Magnitude of best-fit coefficients for each case. The top row
shows CL and the bottom row shows Lnorm. The left column is linearly
scaled and the right column is logarithmically scaled (x -axis only).
measured forces in the current study parallels the constituent parts of that analytical
model.
The force production term for quasi-steady circulatory lift is associated
with meeting the Kutta criteria under the instantaneous conditions. In the von
Ka´rma´n and Sears model, this reduces to the simple thin-airfoil theory solution of
L = CL
1
2
ρU2∞S and CL = 2pi sin(α). This contribution to force production varies
92
proportionally with dynamic pressure and with some expression of force coefficient
and is therefore connected to the C1 component of the current decomposition. The
small-angle approximations inherent in the analysis of von Ka´rma´n and Sears mean
that CL ≈ CN and no distinction is made. For the larger perturbations of interest
in the current work, these force orientations are distinct and the thin-airfoil theory
solution may be more usefully posed as CN = 2pi sin(α). Alternately, the equation
for CN may be altogether replaced by a look-up table based on empirical normal
force coefficients measured under truly steady conditions, which would indirectly
remove some of the flow assumptions (attached, inviscid flow for example) built into
the thin-airfoil theory model.
The second term of von Ka´rma´n and Sears is the inertial force produced by
“apparent mass,” which is connected to the C2 component of the current decomposition.
For small perturbations, this is represented by a potential flow solution of an
accelerating flat plate, CN =
pic ˙V⊥
2U2∞
. This solution has been used successfully in several
studies of large, unsteady events despite the theoretical limitations of the inviscid
flow model [26].
The third term of von Ka´rma´n and Sears is the lift deficiency due to wake
vorticity. This term is not directly included in the current decomposition approach,
although it is unreasonable — particularly for the aggressive plunge maneuvers —
to assume that wake vorticity is insignificant. Some of this wake vorticity force
production contribution will not match either of the decomposition’s constituent
shapes. This would be captured in the residual term, R, which was relatively small
for all cases measured in the current work. Alternately, some of this wake vorticity
93
force contribution will match the shape of the C1 or C2 terms. It is important to
consider this wake vorticity force production term when comparing the observed and
predicted amplitudes for each of the two directly included terms.
The coefficient C1 in the current model corresponds to the magnitude of forces
produced by the relative fluid velocity and so can be compared to simple aerodynamic
models such as thin-airfoil theory or a quasi-steady empirical method.
CN(t) = C˜N(α+θ(t))
(
1+
V (t)2
U2∞
)
(4.11)
where the tilde indicates quasi-steady quantities related to the total flow velocity
rather than the pre-maneuver freestream velocity. The angle α is the geometric pitch
of the wing relative to horizontal, while θ = tan
(
V
U∞
)−1
is the relative plunge velocity
angle. Thin airfoil theory (with an aspect-ratio correction) predicts C˜N =
2pi
1+ 2
A
sin(θ),
while the empirical look-up method returns values from the static force data.
These equations represent the velocity-proportional force production, but do
not share the exact form of C1V (t)2 which would be needed for a direct comparison
to the decomposed measured forces. Both thin-airfoil theory and static data give
values for C˜N which vary as a function of θ(t). The predicted circulatory lift histories
(for example, the orange line in figures 4.8 and 4.9) are not exactly sinusoidal, and
so the peak value is not truly representative of C1. To facilitate the comparison
between observed and predicted circulatory forces, the predicted force histories
were also characterized by a least-squares fit of C1V (t)2 so that the C1 values from
measurements and predictions could be directly compared in figure 4.11.
94
0 10 20 30 40 50 60 70 80 90
0
1
2
3
4
5
6
7
8
9
10 C1 from measurements
thin-airfoil theory
quasi-steady empirical
Figure 4.11: Comparison of measured and predicted lift for velocity-
proportional C1 term
Figure 4.11 shows the C1 values from force measurements (blue triangles)
normalized by mean kinetic energy and plotted as a function of peak effective angle
of attack, θ = tan
(
2h
w
)−1
. Variations in force production with respect to peak
dynamic pressure are collapsed by the normalization, so the remaining variations
are primarily a function of the plunge-velocity-induced flow incidence angle — or, in
the case of the analytical models, exclusively so. Thin-airfoil theory, plotted as the
solid line in figure 4.11, accurately predicts lift for gentle plunge maneuvers which
reach peak effective angles of attack below 45°. The treatment of flow over the wing
as inviscid and therefore attached leads to over-prediction of force production at
excessively high peak incidence angles. Viscosity and flow separation can be indirectly
incorporated into the aerodynamic model by using an empirical look-up table. The
95
results of this semi-empirical method are plotted as the dashed line in figure 4.11.
In general, this model under-predicts the plunge maneuver lift production. The C1
values from measurements fall somewhere between these two predictions, suggesting
that the wake vorticity effects — the here-unaccounted-for term from von Ka´rma´n
and Sears — augment the transient lift but not as much as though the flow was
entirely attached. As was the case in the model by Babinsky et al., the transient
lift is more closely approximated by attached flow than by separated flow (equation
1.18). It is possible that the influence of a substantial leading edge vortex could
justify use of the attached flow thin-airfoil theory results at angles well beyond stall
[28].
The coefficient C2 represents the magnitude of forces produced by the fluid
acceleration. The potential flow approximation of inertial forces in equation 1.6
applied to the sine-squared plunge maneuver at zero geometric angle of attack
produces:
Faccel =
ρpi2c2AhU2∞
w2
sin
(
pit∗
w
)
cos
(
pit∗
w
)
(4.12)
∴ C2 =
2pi2h
w2
(
1
2
ρU2∞c
2
A
)
(4.13)
This predicted amplitude can be easily compared to the C2 values from measurements.
As before, figure 4.12 shows a scatter of the best-fit C2 from measurements plotted
against an x-axis for which the available model collapses to a single line. In this
case, that x-axis parameter is less intuitive; it is found by normalizing the prediction
for added-mass force, equation 1.3, using the kinetic-energy method, equation 2.5.
This step ensures that the model for predicting C2, equation 4.13, can be visualized
96
0 1 2 3 4 5 6
0
2
4
6
8
10
12
14
16
18
20
30
40
50
60
70
80
C2 from measurements
potential flow model
linear fit
Figure 4.12: Comparison of measured and predicted lift for acceleration-
proportional C2 term
as a single line in figure 4.12.
The potential flow model for added-mass, plotted as the solid line in figure
4.12, under-predicts the inertial force amplitudes, C2, that were extracted from the
measurements in most cases. Interestingly, most cases were under-predicted by nearly
the same factor, and so a semi-empirical model can be formulated which greatly
improves the agreement between predictions and measurements:
C′2 = 1.6C2 (4.14)
This empirically tuned potential flow model is plotted as the dashed line in figure
4.12.
97
The color of each C2 point in figure 4.12 corresponds to the peak effective
angle of attack reached during that maneuver. It is apparent that the points which
agree with one or the other predicted line are grouped by this peak angle. Gentle
plunge maneuvers which reached peak θ < 40° are closer to the potential flow model
while aggressive plunge maneuvers which reached peak θ > 50° are closer to the
linear fit. This trend indicates that some unsteady effect may be contributing to
the C2 component of lift production. This increase in force production may also
be attributed to wake vorticity, although the mechanism is less clear than in the
interpretation of observed C1 magnitudes.
In combination, these terms form a model which accurately represents force
production from somewhat low-amplitude sine-squared plunge maneuvers. The
quasi-steady model based on thin-airfoil theory is:
CN(t) =
2pi
1+ 2
A
sin(θ(t))
(
1+
V (t)2
U2∞
)
+
pic ˙V (t)
2U2∞
(4.15)
where again, θ(t) = tan
(
V(t)
U∞
)−1
. Each term of this model matched the observed
amplitudes for cases in which θ ≤ 45° (see figures 4.11 and 4.12). Cases in which
θ > 45° are over-predicted by the velocity-proportional term, C1, and under-predicted
by the acceleration-proportional term, C2. The acceleration term can be seen to
converge to a new value for those higher angles, and so in those cases it may be
useful — if not physically insightful — to “tune” the model by scaling the C2 term
by 1.6.
Additionally, the observations made in section 4.2.4 indicate that the wing
98
pitch angle has little effect on the unsteady force production for most cases tested in
the current work. Although the effects of wing pitch angle on the unsteady force
were observed to be negligible for most of the tested kinematics, those effects can
still be incorporated into the model so that the steady-state predictions better match
to measured forces and the inertial force term better matches to the potential flow
model. The acceleration-proportional term of equation 4.15 can be made to match
that of the potential flow added-mass model by scaling based on the plate-normal
projection using a factor of cos(α). The effects of the wing pitch angle on the
velocity-proportional term of equation 4.15 only appeared to affect the steady-state
forces, which can be represented by an additional term, C˜N(α). Each of these effects
is small compared to the total unsteady force, but they are still included in the final
model:
CN(t) = C˜N(α) +
2pi
1+ 2
A
sin(θ(t))
(
1+
V (t)2
U2∞
)
+
pic ˙V (t)
2U2∞
cos(α) (4.16)
This model presents the normal-force coefficient resulting from a sine-squared profile
plunge maneuver as the sum of three parts: the empirical steady-state coefficient
due to the wing pitch angle, the transient circulatory force linked to the leading edge
vortex, and the non-circulatory inertial force. This is a semi-empirical quasi-steady
model which was made to match unsteady force production by strictly limiting the
plunge maneuver to sine-squared kinematics. Note that the inertial component of
the force coefficient is a function of wing chord, c, which indicates that the relative
contribution of inertial and circulatory forces do not scale equally. For geometrically-
99
0 0.5 1 1.5
-6
-4
-2
0
2
4
6
8
10
12
Measured force
Predicted force
Tuned inertial
Figure 4.13: Comparison of measured and predicted force production for a
high velocity ratio plunge maneuver: w = 1c, h = 1c, α = 0°.
similar plunge maneuvers, wings of unequal size will produce the same normalized
circulatory force (that is the purpose of the normalization), but different inertial
forces.
Figures 4.13, 4.14, and 4.15 show examples of plunging wing force history
predictions made by this model compared to the measured forces. As expected based
on the discussion of C1 and C2 amplitudes, this model accurately represents forces
produced by low-amplitude plunge maneuvers, but is increasingly incorrect as the
velocity ratio increases. For the high-amplitude plunge maneuver in figure 4.13, the
force production is mainly inertial, and the empirically tuned C2 term improves the
model accuracy. Figure 4.14 shows results for a low-amplitude plunge maneuver.
In this case (and all similar cases) the model matches the measured forces closely.
100
-2 0 2 4 6 8 10
-1
0
1
2
3
4
5
Measured force
Predicted force
Tuned inertial
Figure 4.14: Comparison of measured and predicted force production for a
low velocity ratio plunge maneuver: w = 6c, h = 3c, α = 0°.
Both of these figures show results from cases for which α = 0°. Figure 4.15 shows
results from a high-amplitude plunge maneuver for which α = 20°. The substantial
unsteady force production dominates, so the steady-state influence of the wing pitch
angle does not visibly affect the results. In this case, θ ≈ 80° and so the model
over-predicts the force production as expected based on figure 4.11.
101
0 0.5 1 1.5
-6
-4
-2
0
2
4
6
8
10
12
Measured force
Predicted force
Tuned inertial
Figure 4.15: Comparison of measured and predicted force production for a
high velocity ratio plunge maneuver with wing pitch angle: w = 1c, h = 3c,
α = 20°.
4.4 Comparison to gusts
It is also useful to compare the force production between the transverse gust
encounters and the sine-squared plunge maneuvers, which are geometrically similar
moving model and moving wing scenarios. In both cases, the steady flow is horizontal
and the transient flow is vertical and has a sine-squared velocity profile.
The plunge maneuver kinematics are simpler to create and vary than is the
generated fluid gust profile [46] (this is why even in the current work there are many
more plunge cases than gust encounters — compare tables 2.1 and 2.2). Because of
this, the investigation of gust encounter force production would be greatly simplified if
102
-1 0 1 2 3 4 5 6
-1
0
1
2
3
4
Plunge forces
Plunge inertial estimate
Plunge minus inertial
 Distributed
Gust forces
Figure 4.16: Comparison of normal force histories for a set of geometrically
similar gust encounter and plunge maneuver at α = 0°.
the results from plunging wings could be used to represent the gust encounter and so
remove the need for a fluid gust generator. The current work includes measurements
of both scenarios, and so is able to compare the force production in each and identify
the inherent differences.
Figures 4.16 and 4.17 show force histories from gust encounters and plunge
maneuvers as closely matched as was available in the data of the current work. The
forces produced during the gust encounters are plotted in blue, and those from the
plunge maneuvers are in red. The other lines are discussed in the following sections.
In figure 4.16, the gust encounter and plunge maneuver both take place at α = 0°,
but the widths and velocity ratios are not exactly matched. The gust width is fixed
at w = 3.2c, and in this case the velocity ratio was G = Vg
U∞ = 0.42. The closest
103
-1 0 1 2 3 4 5 6
-1
0
1
2
3
4
Plunge forces
Plunge inertial estimate
Plunge minus inertial
 Distributed
Gust forces
Figure 4.17: Comparison of normal force histories for a set of geometrically
similar gust encounter and plunge maneuver at α = 20°.
plunge maneuver had a width of w = 3c and a velocity ratio of Vp
U∞ = 0.67. In figure
4.17, the gust encounter and plunge maneuver both take place at α = 20°, but the
gust encounter was defined by w = 3.2c, G = 0.84, and the closest plunge maneuver
was w = 3c, Vp
U∞ = 0.67.
It is apparent and expected that the force production from similar-kinematics
gust encounters and plunge maneuvers are not inherently the same. The differences
between these cases must be quantified and counteracted in order to most closely
match the force histories. In this work, no process is found which allows for the
complete replacement of results from gust encounters by results from plunging wings,
although the progress towards this goal is still worth discussing.
104
4.4.1 Inertial forces
The most obvious difference between moving model and moving fluid scenarios
(for kinematics which include acceleration) is the presence or lack of inertial forces.
In the gust encounters in the current work, the fluid is not accelerating — instead,
the wing is passing through a region of varying fluid velocity. This does not induce
any inertial force production. Alternately, the plunging wing experiences significant
inertial forces during its acceleration. While the inertial response caused by the mass
of the wing (and measurement hardware) was removed during post-processing, there
is still an inertial force contribution from the surrounding fluid, the “added-mass”
force.
Because this quantity is present in only the plunge maneuver cases, it can
simply be subtracted from the plunge force histories. The magnitude of the added-
mass force is given by equation 1.3, which reduces here to the right-most term of
equation 4.16. This is plotted in figures 4.16 and 4.17 as the green line. The result of
subtracting this from the total plunge force production is the “non-inertial” plunge
force production, plotted in purple. This procedure has not caused the gust encounter
and plunge maneuver forces to coincide, but it has caused the plunge maneuver
force histories to be similar in shape to the gust encounter force histories, rising
from steady state to peak over t∗ ≈ 1.5 and then descending to steady state without
overshoot. As was also observed in the investigation of harmonic plunging by Wong
et al. [46], the difference in forces produced by the moving model and moving fluid
cases at the moment of zero acceleration indicates that inertial effects are not the
105
only difference between transverse gusts and plunge maneuvers.
4.4.2 Velocity distribution
The more complex difference between the gust encounters and the plunge
maneuvers is the chord-wise distribution of the transient velocity. In the case of the
plunge maneuvers, the kinematics vary in time, so the leading and trailing edges of
the wing (and all points in between) experience simultaneous variations in transient
velocity. This is not the same in the gust encounters. The gust profile is fixed in
space, and the wing passes steadily through it. The leading edge of the wing reaches
the gust first, and each point along the wing experiences different transient velocities
than one another.
One effect of this is that the trailing edge of the wing does not exit the gust
until t∗ = w+ c cos(α), making the gust encounter last up to one chord-length longer
than the width of the fluid gust. However, the forces observed for the gust encounters
in the current work were most closely related to the conditions at the leading edge of
the wing. Because of this, the gust encounters were compared to plunge maneuvers
with widths of w = 3c rather than w = 4c.
Additionally, there is no moment during the gust encounter during which the
entire wing experiences the peak gust velocity. This means that the chord-averaged
peak dynamic pressure is lower in the gust encounter than in a plunge maneuver
with the same characteristic velocity. This can be compensated for by effectively
distributing the plunge maneuver velocity profile in the same way that the gust
106
velocity is distributed. The velocity itself can not be physically distributed in
the experiment; instead, the force production resulting from that velocity can be
distributed in the analysis.
At any moment during the gust encounter, each chord-wise location on the wing
experiences a different nominal gust velocity. During the plunge maneuver, however,
each location experiences the same simultaneous plunge velocity. The effects of
the plunge can be distributed over the velocity profile of the gust by treating each
chord-wise location on the wing as producing the plunge forces from the moment
during which the wing was plunging at the appropriate velocity. For example, at
t∗ = 0.5 the trailing half of the wing has not entered the gust, and so in the plunge
case no force contribution is considered there, even though at t∗ = 0.5 the entire
wing was plunging. The force production on each chord-wise point of the plunging
wing is taken from the plunge force history at the moment that matches the gust
velocity at that time and chord-wise location.
The distributed force production at any point is the average over the preceding
chord-length of the un-distributed force production (accounting for the horizontal
projection of the chord, the average is taken over t∗ = cos(α)). This procedure
incorporates both the effective delay in the transient event for the trailing edge
compared to the leading edge and the reduced peak chord-averaged transient velocity.
The result of this is plotted as the dashed purple line in figures 4.16 and 4.17. In
these cases, this distribution of the plunge maneuver force production does not make
it conclusively closer or further from the gust encounter force production. The effects
of the velocity distribution — and thus the ability to test this procedure — would
107
be increased for a narrower gust encounter (or larger wing), but that is not available
to the current study.
4.5 Chapter summary
In this chapter, force measurements from sine-squared profile plunge maneuvers
were displayed and analyzed. The force histories were shown to be dominated by
plate-normal forces, to vary significantly in shape depending on the plunge kinematics,
and to recover to steady state force production almost immediately after the plunge
maneuver ended.
Variations in force production were then inspected with respect to one
independent parameter at a time. The freestream velocity had little effect on the
normalized force histories, which indicates that the force production is insensitive to
Reynolds number within the tested range. The wing pitch angle also had little effect
on the transient force production, which was unexpected and remains unexplained.
The plunge maneuver width and height both changed the force histories, because
they each changed the ratio of steady and transient velocities. The measured data
was reduced to characteristic scalars by curve fitting based on the instantaneous
acceleration and dynamic pressure of the maneuver. The reduced data helped to
identify that forces predicted by thin-airfoil theory and potential flow added-mass
matched the measured force production closely for roughly half of the tested cases.
The other half of the tested cases, which had higher velocity ratios, were over-
predicted by thin-airfoil theory and under-predicted by added-mass, which likely
108
indicates that unmodeled unsteady force production becomes more significant in
those cases. The resulting force production model was presented (equation 4.16)
and the predicted and measured force histories were compared for several example
kinematics.
Force production was then compared between the plunge maneuvers and the
gust encounters. Two important differences, inertial forces and spatial velocity
distribution, were discussed and methods of counteracting those differences were
tested. For the available data, the described treatment of force histories was not
adequate to collapse results between moving model and moving fluid force production.
109
Chapter 5: Summary and conclusions
An experimental investigation of unsteady force production by stream-normal
phenomena was conducted in order to extend present unsteady force models to these
two-dimensional, large-perturbation conditions. Consideration was given to small-
perturbation analyses of two-dimensional events including Wagner [10], Ku¨ssner
[12], and von Ka´rma´n and Sears [18], and to large-perturbation analysis of quasi-
one-dimensional events such as Granlund et al. [33] and Babinsky et al. [26]. The
current work adapted the existing small-perturbation models to large-velocity-ratio
transverse gust encounters and plunge maneuvers by the physical interpretation
of correlations between the defined kinematic parameters and the observed force
production.
Unsteady force production was measured for a flat plate wing during many
variations of transverse gust encounters and plunge maneuvers in the University of
Maryland automated motion-controlled towing tank. The velocity profile in each
case was limited to the sine-squared shape, but parameters such as the wing pitch
angle, wing towing velocity, gust velocity ratio, plunge maneuver height, and plunge
maneuver width were varied. Flow velocities around the wing were measured for
several gust encounters to aid in identifying the sources of unsteady force fluctuations.
110
Force histories which were measured during gust encounters were characterized
by peak normal force in each case. The magnitudes of these peaks within the
current parameter space were found to be reasonably approximated by quasi-steady
analysis, but the variations in peak force with respect to wing pitch angle suggested
non-negligible unsteady effects (as did the duration of the gust recovery process).
The flow velocity measurements made by PIV identified large leading and trailing
edge vortices as a likely mechanism for the unsteady influence on force production.
Force histories which were measured during the plunge maneuvers were
compared by decomposing the force production into two sinusoidal terms which
corresponded to the quasi-steady terms of the model by von Ka´rma´n and Sears. The
residual of this model was found to be low in all cases, which indicated that unsteady
force production during the tested plunge maneuvers was accurately represented by
quasi-steady analysis. The unsteady influence on force histories instead augmented
each of the quasi-steady terms, which were found to be under-predicted by typical
analytical models.
5.1 Key gust encounter conclusions
The transverse gust encounters in the current work were shown to produce
one primary peak in force production during the gust encounter and diminishing
oscillations in force production after the gust encounter. The gradual return of forces
to the steady-state levels occurred over roughly 30c — an order of magnitude greater
than the duration of the gust encounter.
111
The initial minimum in these oscillations, which occurs just as the trailing
edge of the wing exits the gust, causes the force production to temporarily dip below
the steady-state force production of the towed wing. The force oscillations during
gust recovery were connected to the presence of large vortices in the wake of the
wing. Concentrations of negative vorticity from the leading edge vortices caused force
augmentation, while concentrations of positive vorticity from trailing edge vortices
caused force reduction.
The magnitude of the peak force production in most of the studied gust
encounters was found to match in magnitude to predictions made by quasi-steady
empirical analysis. The trend in peak force production with respect to wing pitch
angle was more accurately recreated by indicial convolution of Ku¨ssner’s function
and non-linear thin airfoil theory, but that model poorly recreated the trend in peak
force production with respect to gust velocity ratio. This close agreement suggests
that Ku¨ssner’s function and non-linear thin airfoil theory adequately represent the
transient force production during these transverse gust encounters but that the
current implementation misses a variation in magnitude which scales with gust ratio
— perhaps unidentified in the current work due to the lack of tested variations in gust
width.
5.2 Key plunge maneuver conclusions
Force production by the sine-squared profile plunge maneuvers in the current
work was shown to vary with respect to the plunge maneuver width and height, but
112
not with respect to freestream Reynolds number and only slightly with respect to
wing pitch angle. Wing pitch angle does significantly affect the steady-state force
production, but the unsteady force production dominates in most of the tested cases.
The observed invariance in transient force production with respect to wing pitch angle
goes against available analytical models and remains unexplained. Although the
unsteady force production was significant during the maneuver, the forces returned
to steady-state levels nearly immediately after the maneuver ended.
A physically-based, quasi-steady model was formulated which matched to the
measured force histories for many of the conditions tested in the current work. The
model combined a semi-empirical steady-state term, the potential flow solution for
added-mass, and quasi-steady thin-airfoil theory. The model did not match as closely
to force histories from plunge maneuvers with very high velocity ratios. Some of
those cases were able to be corrected by an empirical tuning of the inertial force
production estimate.
Fundamental differences between transverse gust encounters and plunge
maneuvers were unable to be counteracted in a way which could allow the translation
of results from one case to the other. Force production by the gust encounters
was significantly lower than that of geometrically-similar plunge maneuvers. This
difference was not adequately accounted for by inertial force production or spatial
velocity distributions.
113
5.3 Highlighted original contributions
In the current work, it was demonstrated that a large-perturbation adaptation
to indicial theory using non-linear thin-airfoil theory and Ku¨ssner’s function was able
to closely match the trends in peak force production by transverse gust encounters for
variations in wing pitch angle. This improved on the predicted trends of both quasi-
steady analysis and the typical application of indicial theory using linearized thin
airfoil theory, although those two models each captured the trends with respect to
gust velocity ratio better than the proposed model. This model extends the preceding
work in small-perturbation transverse gusts and large-perturbation longitudinal gusts
to the current subject of large-amplitude transverse gusts.
It was also demonstrated that the quasi-steady components of the model by von
Ka´rma´n and Sears can account for most of the force production during sine-squared
profile plunge maneuvers. The missing term, which accounts for wake vorticity, may
explain the under-prediction of semi-empirical methods. This agrees with the gust
encounter results of the current work and with preceding work in surge maneuvers,
in which the effect of the leading edge vortex was to augment the force production
so that it approached that of attached flow even at high incidence angles [26].
5.4 Suggestions for future work
The experimental data in the current work and the models developed based
on that data have several limitations which can be addressed in future research.
114
Hardware limitations meant that the fluid gust velocity was not variable during these
tests; the gust velocity ratio was changed by varying the towing velocity of the wing.
It was expected based on previous work that the tested variations in freestream
Reynolds number did not have significant independent effects on the force coefficient
histories, but verification of that assumption is only possible using a gust generator
which has variable speed control.
Additionally, the chord-normalized gust width was not varied in the current
work. This could be varied in future work by a controllable-width gust generator or,
perhaps more easily, by repeating tests in a fixed-width gust using a smaller or larger
wing. This would be particularly helpful in the further development of indicial models,
which integrate the force production over the flow history and so are significantly
changed by the duration (in convective time) of the transient gust velocity. The
investigation of this effect could help to reconcile the otherwise-successful predictions
of the proposed adaptation to indicial theory with the observed trends in peak force
production with respect to gust ratio.
The lack of variation in plunge maneuver force production with respect to wing
pitch angle remains unexplained. The wing pitch angle affects the steady-state force
production and alters the orientation of the plate-normal force, but it had little effect
on the magnitude of the transient plate-normal force. For most of the cases tested in
the current work, the transient forces were far greater than the steady-state forces,
so the wing pitch angle appeared to have little overall effect on the force production
during the plunge maneuvers. Neither thin-airfoil theory nor the empirical look-up
method agree with the observed result that the variation in normal force production
115
with respect to the plunge-velocity-induced flow incidence angle, dCN
dθ
, is unaffected
by the initial pitch angle, α. Further investigation of the effect of wing pitch angle
on plunge maneuver force production is needed to identify the mechanisms behind
the observations of the current work.
During the decomposition of plunge maneuver force production, the model
fit residual was low for all cases despite the conflicting nature of applying a quasi-
steady model to highly unsteady force production. The effects of the unsteady
wake vorticity term of the model by von Ka´rma´n and Sears were thought to be
contained in the quasi-steady model terms, rather than being truly absent as the low
residual might suggest. This was helpful in the development of a physically-based
quasi-steady model for plunging wing force production, but limits that result to
sine-squared profile plunge maneuvers. Further analysis of von Ka´rma´n and Sears for
large-perturbation plunge maneuvers could separate the unsteady and quasi-steady
effects by testing maneuvers with non-sinusoidal velocity profiles, for which the quasi-
steady and unsteady force production components would have more notably different
shapes. This would significantly increase the parameter space and complicate the
decomposition method of analysis but could eventually allow for a more generalized
model for unsteady force production during plunge maneuvers with separated flow.
The vortex dynamics of a surging flat plate were simplified by Babinsky et al.
so that their unsteady effects could be expressed in a quasi-steady model [24–26].
The constant longitudinal acceleration of that maneuver greatly simplified the vortex
trajectories compared to what could be expected for the plunging wing maneuvers
of the current work. The two-dimensional relative motion of the wing and the
116
shed vortices in the plunging case would require a more elaborate characterization
than the one-dimensional motion of the surging wing. Despite this difficulty, PIV
measurements of vortex growth and trajectories during plunging maneuvers could
provide an alternate formulation for a quasi-steady model of plunging wing force
production. A similar approach could be applied to transverse gusts, although the
vortex trajectories may be even further complicated due to the leading edge vortices
being temporarily “shielded” from the gust by the wing. Even if vortex tracking
does not lead to quasi-steady models for force production in these cases, increased
knowledge of the vortex behavior for various transverse gusts and plunging maneuvers
may help to justify the observed forces in the current work.
117
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