# Flow-induced oscillations of pitching swept wings: Stability boundary,
vortex dynamics and force partitioning

Yuanhang Zhu1<EMAIL_ADDRESS>Kenneth Breuer1  1 Center for Fluid
Mechanics, School of Engineering, Brown University, Providence, RI 02912, USA

###### Abstract

We experimentally study the aeroelastic instability boundaries and three-
dimensional vortex dynamics of pitching swept wings, with the sweep angle
ranging from 0 to 25 degrees. The structural dynamics of the wings are
simulated using a cyber-physical control system. With a constant flow speed, a
prescribed high inertia and a small structural damping, we show that the
system undergoes a subcritical Hopf bifurcation to large-amplitude limit-cycle
oscillations (LCOs) for all the sweep angles. The onset of LCOs depends
largely on the static characteristics of the wing. The saddle-node point is
found to change non-monotonically with the sweep angle, which we attribute to
the non-monotonic power transfer between the ambient fluid and the elastic
mount. An optimal sweep angle is observed to enhance the power extraction
performance and thus promote LCOs and destabilize the aeroelastic system. The
frequency response of the system reveals a structural-hydrodynamic oscillation
mode for wings with relatively high sweep angles. Force, moment, and three-
dimensional flow structures measured using multi-layer stereoscopic particle
image velocimetry are analyzed to explain the differences in power extraction
for different swept wings. Finally, we employ a physics-based Force and Moment
Partitioning Method (FMPM) to quantitatively correlate the three-dimensional
vortex dynamics with the resultant unsteady aerodynamic moment.

###### keywords:

flow–structure interactions, vortex dynamics

##  1 Introduction

The fluid-structure interaction (FSI) of elastically mounted pitching wings
can lead to large-amplitude flow-induced oscillations under certain operating
conditions. In extreme cases, these flow-induced oscillations may affect
structural integrity and even cause catastrophic aeroelastic failures (Dowell
et al., 1989). On the other hand, however, hydro-kinetic energy can be
harnessed from these oscillations, providing an alternative solution for next-
generation renewable energy devices (Xiao & Zhu, 2014; Young et al., 2014;
Boudreau et al., 2018; Su & Breuer, 2019). Moreover, the aero-/hydro-elastic
interactions of passively pitching wings/fins have important connections with
animal flight (Wang, 2005; Bergou et al., 2007; Beatus & Cohen, 2015; Wu et
al., 2019) and swimming (Long & Nipper, 1996; Quinn & Lauder, 2021), and
understanding these interactions may further aid the design and development of
flapping-wing micro air vehicles (MAVs) (Shyy et al., 2010; Jafferis et al.,
2019) and oscillating-foil autonomous underwater vehicles (AUVs) (Zhong et
al., 2021b; Tong et al., 2022).

Flow-induced oscillations of pitching wings originate from the two-way
coupling between the structural dynamics of the elastic mount and the fluid
force exerted on the wing. While the dynamics of the elastic mount can be
approximated by a simple spring-mass-damper model, the fluid forcing term is
usually found to be highly nonlinear due to the formation, growth, and
shedding of a strong leading-edge vortex (LEV) (McCroskey, 1982; Dimitriadis &
Li, 2009; Mulleners & Raffel, 2012; Eldredge & Jones, 2019). Onoue et al.
(2015) and Onoue & Breuer (2016) experimentally studied the flow-induced
oscillations of a pitching plate whose structural stiffness, damping and
inertia were defined using a cyber-physical system (§2.1, see also Hover et
al. (1997); Mackowski & Williamson (2011); Zhu et al. (2020)) and, using this
approach, identified a subcritical bifurcation to aeroelastic instability. The
temporal evolution of the LEV associated with the aeroelastic oscillations was
characterized using particle image velocimetry (PIV), and the unsteady flow
structures were correlated with the unsteady aerodynamic moments using a
potential flow model. Menon & Mittal (2019) numerically studied a similar
problem, simulating an elastically mounted two-dimensional NACA-0015 airfoil
at a Reynolds number of 1000. An energy approach, which bridges prescribed
sinusoidal oscillations and passive flow-induced oscillations, was employed to
characterize the dynamics of the aeroelastic system. The energy approach maps
out the energy transfer between the ambient flow and the elastic mount over a
range of prescribed pitching amplitudes and frequencies and unveils the system
stability based on the sign of the energy gradient.

More recently, Zhu et al. (2020) characterized the effect of wing inertia on
the flow-induced oscillations of pitching wings and the corresponding LEV
dynamics. Two distinct oscillation modes were reported: (i) a structural mode,
which occurred via a subcritical bifurcation and was associated with a high-
inertia wing, and (ii) a hydrodynamic mode, which occurred via a supercritical
bifurcation and was associated with a low-inertia wing. The wing was found to
shed one strong LEV during each half-pitching cycle for the hydrodynamic mode,
whereas a weak secondary LEV was also shed in the high-inertial structural
mode.

These previous studies have collectively demonstrated that LEV dynamics play
an important role in shaping flow-induced oscillations and thus regulate the
stability characteristics of passively pitching wings. However, these studies
have only focused on studying the structural and flow dynamics of two-
dimensional wings or airfoils. The extent to which these important findings
for two-dimensional wings hold in three dimensions remains unclear.

Swept wings are commonly seen for flapping-wing fliers and swimmers in nature
(Ellington et al., 1996; Lentink et al., 2007; Borazjani & Daghooghi, 2013;
Bottom II et al., 2016; Zurman-Nasution et al., 2021), as well as on many
engineered fixed-wing flying vehicles. It is argued that wing sweep can
enhance lift generation for flapping wings because it stabilizes the LEV by
maintaining its size through spanwise vorticity transport – a mechanism
similar to the lift enhancement mechanism of delta wings (Polhamus, 1971).
Chiereghin et al. (2020) found significant spanwise flow for a high-aspect
ratio plunging swept wing at a sweep angle of 40 degrees. In another study,
for the same sweep angle, attached LEVs and vortex breakdown were observed
just like those on delta wings (Gursul & Cleaver, 2019). Recent works have
shown that the effect of wing sweep on LEV dynamics depends strongly on wing
kinematics. Beem et al. (2012) showed experimentally that for a plunging swept
wing, the strong spanwise flow induced by the wing sweep is not sufficient for
LEV stabilization. Wong et al. (2013) reinforced this argument by comparing
the LEV stability of plunging and flapping swept wings and showed that two-
dimensional (i.e. uniform without any velocity gradient) spanwise flow alone
cannot stabilize LEVs – there must be spanwise gradients in vorticity or
spanwise flow so that vorticity can be convected or stretched. Wong & Rival
(2015) demonstrated both theoretically and experimentally that the wing sweep
improves relative LEV stability of flapping swept wings by enhancing the
spanwise vorticity convection and stretching so as to keep the LEV size below
a critical shedding threshold (Rival et al., 2014). Onoue & Breuer (2017)
experimentally studied elastically mounted pitching unswept and swept wings
and proposed a universal scaling for the LEV formation time and circulation,
which incorporated the effects of the pitching frequency, the pivot location
and the sweep angle. The vortex circulation was demonstrated to be independent
of the three-dimensional vortex dynamics. In addition, they concluded that the
stability of LEV can be improved by moderating the LEV circulation through
vorticity annihilation, which is largely governed by the shape of the leading-
edge sweep, agreeing with the results of Wojcik & Buchholz (2014). More
recently, Visbal & Garmann (2019) numerically studied the effect of wing sweep
on the dynamic stall of pitching three-dimensional wings and reported that the
wing sweep can modify the LEV structures and change the net aerodynamic
damping of the wing. The effect of wing sweep on the LEV dynamics and
stability, as one can imagine, will further affect the unsteady aerodynamic
forces and thereby the aeroelastic response of pitching swept wings.

Another important flow feature associated with unsteady three-dimensional
wings is the behavior of the tip vortex (TV). Although the tip vortex usually
grows distinctly from the leading-edge vortex for rectangular platforms (Taira
& Colonius, 2009; Kim & Gharib, 2010; Hartloper et al., 2013), studies have
suggested that the TV is able to anchor the LEV in the vicinity of the wing
tip, which delays LEV shedding (Birch & Dickinson, 2001; Hartloper et al.,
2013). Moreover, the tip vortex has also been shown to affect the unsteady
wake dynamics of both unswept and swept wings (Taira & Colonius, 2009; Zhang
et al., 2020a, b; Ribeiro et al., 2022; Son et al., 2022a, b). However, it
remains elusive how the interactions between LEVs and TVs change with the wing
sweep, and more importantly, how this change will in turn affect the response
of aeroelastic systems.

To dissect the effects of complex vortex dynamics associated with unsteady
wings/airfoils, a physics-based Force and Moment Partitioning Method (FMPM)
has been proposed (Quartapelle & Napolitano, 1983; Zhang et al., 2015; Moriche
et al., 2017; Menon & Mittal, 2021a, b, c) (also known as the vortex
force/moment map method (Li & Wu, 2018; Li et al., 2020a)). The method has
attracted attention recently due to its high versatility for analyzing a
variety type of vortex-dominated flows. Under this framework, the Navier-
Stokes equation is projected onto the gradient of an influence potential to
separate the force contributions from the added-mass, vorticity-induced, and
viscous terms. It is particularly useful for analyzing vortex-dominated flows
because the spatial distribution of the vorticity-induced forces can be
visualized, enabling detailed dissections of aerodynamic loads generated by
individual vortical structures. For two-dimensional airfoils, Menon & Mittal
(2021c) applied FMPM and showed that the strain-dominated region surrounding
the rotation-dominated vortices has an important role to play in the
generation of unsteady aerodynamic forces. For three-dimensional wings, this
method has been implemented to study the contributions of spanwise and cross-
span vortices to the lift generation of rectangular wings (Menon et al.,
2022), the vorticity-induced force distributions on forward- and backward-
swept wings at a fixed angle of attack (Zhang & Taira, 2022), and the
aerodynamic forces on delta wings (Li et al., 2020b). More recently, efforts
have been made to apply FMPM to the analysis of experimental data, in
particular, flow fields obtained using particle image velocimetry. Zhu et al.
(2023) employed FMPM to analyze the vortex dynamics of a two-dimensional wing
pitching sinusoidally in a quiescent flow. Several practical issues in
applying FMPM to PIV data were discussed, including the effect of phase-
averaging and potential error sources.

In this study, we apply FMPM to three-dimensional flow field data measured
using three-component PIV, and use the results to gain insight into the three-
dimensional vortex dynamics and the corresponding unsteady forces acting on
elastically mounted pitching swept wings. We extend the methodology developed
in Zhu et al. (2020), and employ a layered stereoscopic PIV technique and the
FMPM to quantify the three-dimensional vortex dynamics. In the following
sections, we first introduce the experimental setup and method of analysis
(§2). The static force and moment coefficients of the wings are measured
(§3.1) before we characterize the amplitude response (§3.2) and the frequency
response (§3.3) of the system. Next, we associate the onset of flow-induced
oscillations with the static characteristics of the wing (§3.4) and use an
energy approach to explain the nonlinear stability boundaries (§3.5). The
unsteady force and moment measurements, together with the three-dimensional
flow structures (§3.6) are then analyzed to explain the differences in power
extraction for unswept and swept wings. Finally, we apply the Force and Moment
Partitioning Method to quantitatively correlate the three-dimensional vortex
dynamics with the resultant unsteady aerodynamic moment (§3.7). All the key
findings are summarized in §4.

##  2 Methods

Figure 1: (_a_) A schematic of the experimental setup. (_b_) Sketches of
unswept and swept wings used in the experiments. The pivot axes are indicated
by black dashed lines. The green panels represent volumes traversed by the
laser sheet for three-dimensional phase-averaged stereoscopic PIV
measurements.

###  2.1 Cyber-physical system and wing geometry

We perform all the experiments in the Brown University free-surface water
tunnel, which has a test section of $W\times D\times L=0.8~{}\mathrm{m}\times
0.6~{}\mathrm{m}\times 4.0~{}\mathrm{m}$. The turbulence intensity in the
water tunnel is around 2% at the velocity range tested in the present study.
Free-stream turbulence plays a critical role in shaping small-amplitude
laminar separation flutter (see Yuan et al. (2015)). However, as we will show
later, the flow-induced oscillations and the flow structures observed in the
present study are of high amplitude and large size, and we do not expect the
free-stream turbulence to play any significant role. Figure 1(_a_) shows a
schematic of the experimental setup. Unswept and swept NACA 0012 wings are
mounted vertically in the tunnel, with an endplate on the top as a symmetry
plane. The wing tip at the bottom does not have an endplate. The wings are
connected to a six-axis force/moment transducer (ATI Delta IP65) via a wing
shaft. The shaft further connects the transducer to an optical encoder (US
Digital E3-2500) and a servo motor (Parker SM233AE) coupled with a gearbox
(SureGear PGCN23-0525).

We implement a cyber-physical system (CPS) to facilitate a wide structural
parameter sweep (i.e. stiffness, $k$, damping, $b$, and inertia, $I$) while
simulating real aeroelastic systems with high fidelity. Details of the CPS
have been discussed in Zhu et al. (2020), therefore, only a brief introduction
will be given here. In the CPS, the force/moment transducer measures the fluid
moment, $M$, and feeds the value to the computer via a data acquisition (DAQ)
board (National Instruments PCIe-6353). This fluid moment is then added to the
stiffness moment ($k\theta$) and the damping moment ($b\dot{\theta}$) obtained
from the previous time step to get the total moment. Next, we divide this
total moment by the desired inertia ($I$) to get the acceleration
($\ddot{\theta}$) at the present time step. This acceleration is then
integrated once to get the velocity ($\dot{\theta}$) and twice to get the
pitching angle ($\theta$). This pitching angle signal is output to the servo
motor via the same DAQ board. The optical encoder, which is independent of the
CPS, is used to measure and verify the pitching angle. At the next time step,
the CPS recalculates the total moment based on the measured fluid moment and
the desired stiffness and damping, and thereby continues the loop.

Our CPS control loop runs at a frequency of 4000 Hz, which is well beyond the
highest Nyquist frequency of the aeroelastic system. Noise in the force/moment
measurements can be a potential issue for the CPS. However, because we are
using a position control loop, where the acceleration is integrated twice to
get the desired position, our system is less susceptive to noise. Therefore,
no filter is used within the CPS control loop. The position control loop also
requires the pitching motor to follow the commanded position signal as closely
as possible. This is achieved by carefully tuning the PID
(Proportional–Integral–Derivative) parameters of the pitching motor. The CPS
does not rely on any additional tunable parameters other than the virtual
inertia, damping, and stiffness. We validate the system using ‘ring-down’
experiments, as shown in the appendix of Zhu et al. (2020). Moreover, as we
will show later, the CPS results match remarkably well with prescribed
experiments (§3.5), demonstrating the robustness of the system.

The unswept and swept wings used in the present study are sketched in figure
1(_b_). All the wings have a span of $s=0.3$ m and a chord length of $c=0.1$
m, which results in a physical aspect ratio of $AR=3$. However, the effective
aspect ratio is 6 due to the existence of the symmetry plane (i.e. the
endplate). The minimum distance between the wing tip and the bottom of the
water tunnel is around $1.5c$. The chord-based Reynolds number is defined as
$Re\equiv\rho U_{\infty}c/\mu$, where $U_{\infty}$ is the free-stream
velocity, $\rho$ and $\mu$ are water density and dynamic viscosity,
respectively. We set the free-stream velocity to be $U_{\infty}=0.5$
$\mathrm{m~{}s^{-1}}$ for all the experiments (except for particle image
velocimetry measurements, see §2.2), which results in a constant Reynolds
number of $Re=50~{}000$, matching the $Re$ used in Zhu et al. (2020) to
facilitate direct comparisons. For both unswept and swept wings, the leading
edge (LE) and the trailing edge (TE) are parallel. Their pivot axes,
represented by vertical dashed lines in the figure, pass through the mid-chord
point $x/c=0.5$ of the mid-span plane $z/s=0.5$. We choose the current
location of the pitching axis because it splits the swept wings into two
equal-area sections (fore and aft). Moving the pitching axis or making it
parallel to the leading edge will presumably result in different system
dynamics, which will be investigated in future studies.

The sweep angle, $\Lambda$, is defined as the angle between the leading edge
and the vertical axis. Five wings with $\Lambda=0^{\circ}$ (unswept wing),
$10^{\circ},15^{\circ},20^{\circ}$ and $25^{\circ}$ (swept wings) are used in
the experiments. Further expanding the range of wing sweep would presumably
bring more interesting fluid-structure interaction behaviors. However, as we
will show in the later sections, there is already a series of rich (nonlinear)
flow physics associated with the current set of unswept and swept wings. Our
selection of the sweep angle is also closely related to the location of the
pitching axis. Currently, the pitching axis passes the mid-chord at the mid-
span. For a $\Lambda=25^{\circ}$ wing, the trailing edge is already in front
of the pitching axis at the wing root, and the leading edge is behind the
pitching axis at the wing tip. Further increasing the sweep angle brings
difficulties in physically pitching the wing for our existing setup.

###  2.2 Multi-layer stereoscopic particle image velocimetry

We use multi-layer phase-averaged stereoscopic particle image velocimetry
(SPIV) to measure the three-dimensional (3D) velocity field around the
pitching wings. We lower the free-stream velocity to $U_{\infty}=0.3$
$\mathrm{m~{}s^{-1}}$ to enable higher temporal measurement resolution. The
chord-based Reynolds number is consequently decreased to $Re=30~{}000$. It has
been shown by Zhu et al. (2020, see their appendix) that the variation of $Re$
in the range of 30 000 – 60 000 does not affect the system dynamics, as long
as the parameters of interest are properly non-dimensionalized. The water flow
is seeded using neutrally buoyant 50 $\mu$m silver-coated hollow ceramic
spheres (Potters Industries) and illuminated using a horizontal laser sheet,
generated by a double-pulse Nd:YAG laser (532 nm, Quantel EverGreen) with a
LaVision laser guiding arm and collimator. Two sCMOS cameras (LaVision,
$2560\times 2160$ pixels) with Scheimpflug adapters (LaVision) and 35mm lenses
(Nikon) are used to capture image pairs of the flow field. These SPIV image
pairs are fed into the LaVision DaVis software (v.10) for velocity vector
calculation using multi-pass cross-correlations (two passes at $64\times 64$
pixels, two passes at $32\times 32$ pixels, both with 50% overlap).

To measure the two-dimensional-three-component (2D3C) velocity field at
different spanwise layers, we use a motorized vertical traverse system with a
range of 120 mm to raise and lower the testing rig (i.e. all the components
connected by the shaft) in the $z$-axis (King et al., 2018; Zhong et al.,
2021a). Due to the limitation of the traversing range, three measurement
volumes (figure 1 _b_ , V1, V2 and V3) are needed to cover the entire wing
span plus the wing tip region. For each measurement volume, the laser sheet is
fixed at the top layer and the rig is traversed upward with a step size of 5
mm. Note that the entire wing stays submerged, even at the highest traversing
position, and for all wing positions, free surface effects are not observed.
The top two layers of V1 are discarded as the laser sheet is too close to the
endplate, which causes reflections. The bottom layer of V1 and the top layer
of V2 overlap with each other. The velocity fields of these two layers are
averaged to smooth the interface between the two volumes. The interface
between V2 and V3 is also smoothed in the same way. For each measurement
layer, we phase-average 1250 instantaneously measured 2D3C velocity fields
over 25 cycles (i.e. 50 measurements per cycle) to eliminate any instantaneous
variations of the flow field while maintaining the key coherent features
across different layers. Finally, 71 layers of 2D3C velocity fields are
stacked together to form a large volume of phase-averaged 3D3C velocity field
($\sim 3c\times 3c\times 3.5c$). The velocity fields of three wing models
($\Lambda=0^{\circ}$, $10^{\circ}$ and $20^{\circ}$) are measured. For the two
swept wings ($\Lambda=10^{\circ}$ and $20^{\circ}$), the laser volumes are
offset horizontally to compensate for the sweep angle (see the bottom
subfigure of figure 1 _b_).

###  2.3 Governing equations and non-dimensional parameters

The one-degree-of-freedom aeroelastic system considered in the present study
has a governing equation

$I\ddot{\theta}+b\dot{\theta}+k\theta=M,$ (1)

where $\theta$, $\dot{\theta}$, and $\ddot{\theta}$ are the angular position,
velocity and acceleration, respectively. $I=I_{p}+I_{v}$ is the effective
inertia, where $I_{p}$ is the physical inertia of the wing and $I_{v}$ is the
virtual inertia that we prescribe with the CPS. Because the friction is
negligible in our system, the effective structural damping, $b$, equals the
virtual damping $b_{v}$ in the CPS. $k$ is the effective torsional stiffness
and it equals the virtual stiffness $k_{v}$. Equation 1 resembles a forced
torsional spring-mass-damper system, where the fluid moment, $M$, acts as a
nonlinear forcing term. Following Onoue et al. (2015) and Zhu et al. (2020),
we normalize the effective inertia, damping, stiffness and the fluid moment
using the fluid inertia force to get the non-dimensional governing equation of
the system:

$I^{*}\ddot{\theta}^{*}+b^{*}\dot{\theta}^{*}+k^{*}\theta^{*}=C_{M},$ (2)

where

$\begin{gathered}\theta^{*}=\theta,~{}\dot{\theta}^{*}=\frac{\dot{\theta}c}{U_{\infty}},~{}\ddot{\theta}^{*}=\frac{\ddot{\theta}c^{2}}{U_{\infty}^{2}},\\\
I^{*}=\frac{I}{0.5\rho c^{4}s},~{}b^{*}=\frac{b}{0.5\rho
U_{\infty}c^{3}s},~{}k^{*}=\frac{k}{0.5\rho
U_{\infty}^{2}c^{2}s},~{}C_{M}=\frac{M}{0.5\rho
U_{\infty}^{2}c^{2}s}.\end{gathered}$ (3)

We should note that the inverse of the non-dimensional stiffness is equivalent
to the Cauchy number, $Ca=1/k^{*}$, and the non-dimensional inertia, $I^{*}$,
is analogous to the mass ratio between the wing and the surrounding fluid. We
define the non-dimensional velocity as $U^{*}=U_{\infty}/(2\pi f_{p}c)$, where
$f_{p}$ is the _measured_ pitching frequency. In addition to the aerodynamic
moment, we also measure the aerodynamic forces that are normal and tangential
to the wing chord, $F_{N}$ and $F_{T}$, respectively. The resultant lift and
drag forces are

$\begin{gathered}L=F_{N}\cos{\theta}-F_{T}\sin{\theta},\\\
D=F_{N}\sin{\theta}+F_{T}\cos{\theta}.\end{gathered}$ (4)

We further normalize the normal force, tangential force, lift and drag to get
the corresponding force coefficients

$C_{N}=\frac{F_{N}}{0.5\rho U_{\infty}^{2}cs},~{}C_{T}=\frac{F_{T}}{0.5\rho
U_{\infty}^{2}cs},~{}C_{L}=\frac{L}{0.5\rho
U_{\infty}^{2}cs},~{}C_{D}=\frac{D}{0.5\rho U_{\infty}^{2}cs}.$ (5)

###  2.4 Force and Moment Partitioning Method

To apply FMPM to three-dimensional PIV data, we first construct an influence
potential that satisfies Laplace’s equation and two different Neumann boundary
conditions on the airfoil and the outer boundary

$\nabla^{2}\phi=0,~{}\text{and}~{}\frac{\partial\phi}{\partial\boldsymbol{\mathrm{n}}}=\begin{cases}[(\boldsymbol{x}-\boldsymbol{x_{p}})\times\boldsymbol{\mathrm{n}}]\cdot\boldsymbol{\mathrm{e_{z}}}&\text{on
airfoil}\\\ 0&\text{on outer boundary}\end{cases},$ (6)

where $\boldsymbol{\mathrm{n}}$ is the unit vector normal to the boundary,
$\boldsymbol{x}-\boldsymbol{x_{p}}$ is the location vector pointing from the
pitching axis $\boldsymbol{x_{p}}$ towards location $\boldsymbol{x}$ on the
airfoil surface, and $\boldsymbol{\mathrm{e_{z}}}$ is the spanwise unit vector
(Menon & Mittal, 2021b). This influence potential quantifies the spatial
influence of any vorticity on the resultant force/moment. It is only a
function of the airfoil geometry and the pitching axis, and does not depend on
the kinematics of the wing. Note that this influence potential should not be
confused with the velocity potential from the potential flow theory. The
boundary conditions of equation 6 are specified for solving the influence
field of the spanwise moment, and they will be different for solving the lift
and drag influence fields. From the three-dimensional velocity data, we can
calculate the $Q$ field (Hunt et al., 1988; Jeong & Hussain, 1995)

$Q=\frac{1}{2}(\|\boldsymbol{\Omega}\|^{2}-\|\boldsymbol{\mathrm{S}}\|^{2}),$
(7)

where $Q$ is the second invariant of the velocity gradient tensor,
$\boldsymbol{\Omega}$ is the vorticity tensor and $\boldsymbol{\mathrm{S}}$ is
the strain-rate tensor. The vorticity-induced moment can be evaluated by

$M_{v}=-2\rho\int_{V}Q\phi~{}\mathrm{d}V,$ (8)

where $\int_{V}$ represents the volume integral within the measurement volume.
The spatial distribution of the vorticity-induced moment near the pitching
wing can thus be represented by the moment density, $-2Q\phi$ (i.e. the moment
distribution field). In the present study, we focus on the vorticity-induced
force (moment) as it has the most important contribution to the overall
unsteady aerodynamic load in vortex-dominated flows. Other forces including
the added-mass force, the force due to viscous diffusion, the forces
associated with irrotational effects and outer domain effects are not
considered although they can be estimated using FMPM as well (Menon & Mittal,
2021b). The contributions from these other forces, along with experimental
errors, might result in a mismatch in the magnitude of the FMPM-estimated
force and force transducer measurements, as shown by Zhu et al. (2023), and
the exact source of this mismatch is under investigation.

##  3 Results and discussion

###  3.1 Static characteristics of unswept and swept wings

Figure 2: (_a_) Static lift coefficient and (_b_) moment coefficient of
unswept and swept wings. Error bars denote standard deviations of the
measurement over 20 seconds.

The static lift and moment coefficient, $C_{L}$ and $C_{M}$, are measured for
the unswept ($\Lambda=0^{\circ}$) and swept wings ($\Lambda=10^{\circ}$ –
$25^{\circ}$) at $Re=50~{}000$ and the results are shown in figure 2. In
figure 2(_a_), we see that the static lift coefficient, $C_{L}(\theta)$, has
the same behavior for all sweep angles, despite some minor variations for
angles of attack higher than the static stall angle $\theta_{s}=12^{\circ}$
(0.21 rad). The collapse of $C_{L}(\theta)$ across different swept wings
agrees with the classic ‘independence principle’ (Jones, 1947) (i.e.
$C_{L}\sim\cos^{2}\Lambda$) at relatively small sweep angles. Figure 2(_b_)
shows that, for any fixed angle of attack, the static moment coefficient,
$C_{M}$, increases with the sweep angle, $\Lambda$. This trend is most
prominent when the angle of attack exceeds the static stall angle. The inset
shows a zoom-in view of the static $C_{M}$ for $\theta=0.14$ – 0.26. It is
seen that the $C_{M}$ curves cluster into two groups, with the unswept wing
($\Lambda=0^{\circ}$) being in Group 2 (G2) and all the other swept wings
($\Lambda=10^{\circ}$ – $25^{\circ}$) being in Group 1 (G1). As we will show
later, this grouping behavior is closely related to the onset of flow-induced
oscillations (§3.2 & §3.4) and it is important for understanding the system
stability. No hysteresis is observed for both static $C_{L}$ and $C_{M}$,
presumably due to free-stream turbulence in the water tunnel.

###  3.2 Subcritical bifurcations to flow-induced oscillations

We conduct bifurcation tests to study the stability boundaries of the
elastically mounted pitching wings. Zhu et al. (2020) have shown that for
unswept wings, the onset of limit-cycle oscillations (LCOs) is independent of
the wing inertia and the bifurcation type (i.e. subcritical or supercritical).
It has also been shown that the extinction of LCOs for subcritical
bifurcations at different wing inertias occurs at a fixed value of the non-
dimensional velocity $U^{*}$. For these reasons, we choose to focus on one
high-inertia case ($I^{*}=10.6$) in the present study. In the experiments, the
free-stream velocity is maintained at $U_{\infty}=0.5$ $\mathrm{m~{}s^{-1}}$.
We fix the structural damping of the system at a small value, $b^{*}=0.13$,
keep the initial angle of attack at zero, and use the Cauchy number, $Ca$, as
the control parameter. To test for the onset of LCOs, we begin the test with a
high-stiffness virtual spring (i.e. low $Ca$) and incrementally increase $Ca$
by decreasing the torsional stiffness, $k^{*}$. We then reverse the operation
to test for the extinction of LCOs and to check for any hysteresis. The
amplitude response of the system, $A$, is measured as the peak absolute
pitching angle (averaged over many pitching cycles). By this definition, $A$
is half of the peak-to-peak amplitude. The divergence angle, $\overline{A}$,
is defined as the mean absolute pitching angle. Although all the divergence
angles are shown to be positive, the wing can diverge to both positive and
negative angles in experiments.

Figure 3: Amplitude response and static divergence for unswept and swept
wings. $\triangleright$: increasing $Ca$, $\triangleleft$: decreasing $Ca$.
The inset illustrates the wing geometry and the pivot axis. The colors of the
wings correspond to the colors of the amplitude and divergence curves in the
figure.

Figure 3 shows the pitching amplitude response and the static divergence angle
for swept wings with $\Lambda=10^{\circ}$ to $25^{\circ}$. Data for the
unswept wing ($\Lambda=0^{\circ}$) are also replotted from Zhu et al. (2020)
for comparison. It can be seen that the system first remains stable without
any noticeable oscillations or divergence (regime 1 in the figure) when $Ca$
is small. In this regime, the high stiffness of the system is able to pull the
system back to a stable fixed point despite any small perturbations. As we
further increase $Ca$, the system diverges to a small static angle, where the
fluid moment is balanced by the virtual spring. This transition is presumably
triggered by free-stream turbulence, and both positive and negative directions
are possible. Due to the existence of random flow disturbances and the
decreasing spring stiffness, some small-amplitude oscillations around the
static divergence angle start to emerge (regime 2). As $Ca$ is further
increased above a critical value (i.e. the Hopf point), the amplitude response
of the system abruptly jumps into large-amplitude self-sustained LCOs and the
static divergence angle drops back to zero, indicating that the oscillations
are symmetric about the zero angle of attack. The large-amplitude LCOs are
observed to be near-sinusoidal and have a dominant characteristic frequency.
After the bifurcation, the amplitude response of the system continues to
increase with $Ca$ (regime 3). We then decrease $Ca$ and find that the large-
amplitude LCOs persist even when $Ca$ is decreased below the Hopf point
(regime 4). Finally, the system drops back to the stable fixed point regime
via a saddle-node (SN) point. A hysteretic bistable region is thus created in
between the Hopf point and the saddle-node point – a hallmark of a subcritical
Hopf bifurcation. In the bistable region, the system features two stable
solutions – a stable fixed point (regime 1) and a stable LCO (regime 4) – as
well as an unstable LCO solution, which is not observable in experiments
(Strogatz, 1994).

We observe that the Hopf points of unswept and swept wings can be roughly
divided into two groups (figure 3, G1 & G2), with the unswept wing
($\Lambda=0^{\circ}$) being in G2 and all the other swept wings
($\Lambda=10^{\circ}$ – $25^{\circ}$) being in G1, which agrees with the trend
observed in figure 2(_b_) for the static moment coefficient. This connection
will be discussed further in §3.4. It is also seen that as the sweep angle
increases, the LCO amplitude at the saddle-node point decreases monotonically.
However, the $Ca$ at which the saddle-node point occurs first extends towards
a lower value ($\Lambda=0^{\circ}\rightarrow 10^{\circ}$) but then moves back
towards a higher $Ca$ ($\Lambda=10^{\circ}\rightarrow 25^{\circ}$). This
indicates that increasing the sweep angle first destabilizes the system from
$\Lambda=0^{\circ}$ to $10^{\circ}$ and then re-stabilizes it from
$\Lambda=10^{\circ}$ to $25^{\circ}$. This non-monotonic behavior of the
saddle-node point will be revisited from a perspective of energy in §3.5. The
pitching amplitude response, $A$, follows a similar non-monotonic trend.
Between $\Lambda=0^{\circ}$ and $10^{\circ}$, $A$ is slightly higher at higher
$Ca$ values for the $\Lambda=10^{\circ}$ wing, whereas between
$\Lambda=10^{\circ}$ and $25^{\circ}$, $A$ decreases monotonically, indicating
that a higher sweep angle is not able to sustain LCOs at higher amplitudes.
The non-monotonic behaviors of the saddle-node point and the LCO amplitude
both suggest that there exists an optimal sweep angle, $\Lambda=10^{\circ}$,
which promotes flow-induced oscillations of pitching swept wings.

###  3.3 Frequency response of the system

Figure 4: (_a_) Frequency response of unswept and swept wings. (_b_ , _c_)
Force decomposition of the structural mode and the structural-hydrodynamic
mode. (_b_) and (_c_) correspond to the filled orange triangle and the filled
green diamond shown in (_a_), respectively. Note that $t/T=0$ corresponds to
$\theta=0$.

The characteristic frequencies of the flow-induced LCOs observed in figure 3
provide us with more information about the driving mechanism of the
oscillations. Figure 4(_a_) shows the measured frequency response,
$f_{p}^{*}$, as a function of the calculated natural (structural) frequency,
$f_{s}^{*}$, and sweep angle. In the figure, $f_{p}^{*}=f_{p}c/U_{\infty}$ and
$f_{s}^{*}=f_{s}c/U_{\infty}$, where $f_{p}$ is the measured pitching
frequency and

$f_{s}=\frac{1}{2\pi}\sqrt{\frac{k}{I}-(\frac{b}{2I})^{2}}$ (9)

is the structural frequency of the system (Rao, 1995). We observe that for all
the wings tested in the experiments and over most of the regimes tested, the
measured pitching frequency, $f_{p}^{*}$, locks onto the calculated structural
frequency, $f_{s}^{*}$, indicating that the oscillations are dominated by the
balance between the structural stiffness and inertia. These oscillations,
therefore, correspond to the _structural_ mode reported by Zhu et al. (2020),
and feature characteristics of high-inertial aeroelastic instabilities. We can
decompose the moments experienced by the wing into the inertial moment,
$I^{*}\ddot{\theta}^{*}$, the structural damping moment,
$b^{*}\dot{\theta}^{*}$, the stiffness moment, $k^{*}\theta^{*}$, and the
fluid moment, $C_{M}$. As an example, for the $\Lambda=10^{\circ}$ wing
pitching at $f^{*}_{s}=0.069$ (i.e. the filled orange triangle in figure 4
_a_), these moments are plotted in figure 4(_b_). We see that for the
structural mode, the stiffness moment is mainly balanced by the inertial
moment, while the structural damping moment and the fluid moment remain
relatively small.

In addition to the structural mode, Zhu et al. (2020) also observed a
hydrodynamic mode, which corresponds to a low-inertia wing. In the
hydrodynamic mode, the oscillations are dominated by the fluid forcing, so
that the measured pitching frequency, $f^{*}_{p}$, stays relatively constant
for a varying $Ca$. In figure 4(_a_), we see that for the $\Lambda=20^{\circ}$
and $25^{\circ}$ wings, $f^{*}_{p}$ flattens near the saddle-node boundary.
This flattening trend shows an emerging fluid-dominated time scale, resembling
a hydrodynamic mode despite the high wing inertia. Taking
$\Lambda=20^{\circ}$, $f^{*}_{s}=0.068$ (i.e. the filled green diamond in
figure 4 _a_) as an example, we can examine the different contributions to the
pitching moments in figure 4(_c_). It is observed that in this oscillation
mode, the stiffness moment balances both the inertial moment and the fluid
moment. This is different from both the structural mode and the hydrodynamic
mode, and for this reason, we define this hybrid oscillation mode as the
_structural-hydrodynamic_ mode.

There are currently no quantitative descriptions of the structural-
hydrodynamic mode. However, it can be qualitatively identified as when the
pitching frequency of a (1:1 lock-in) structural mode flattens as the natural
(structural) frequency increases. Based on the observations in the present
study, we believe this mode is not a fixed fraction of the structural
frequency. Instead, the frequency response shows a mostly flat trend (figure 4
_a_ , green and dark green curves) at high $f_{s}^{*}$, indicating an
increasingly dominating fluid forcing frequency. For a structural mode, the
oscillation frequency locks onto the natural frequency due to the high
inertial moment. However, as the sweep angle increases, the fluid moment also
increases (see also figure 8 _a_). The structural-hydrodynamic mode emerges as
the fluid forcing term starts to dominate in the nonlinear oscillator.

For a fixed structural frequency, $f_{s}^{*}$, as the sweep angle increases,
the measured pitching frequency, $f_{p}^{*}$, deviates from the 1:1 lock-in
curve and moves to lower frequencies. This deviation suggests a growing added-
mass effect, as the pitching frequency $f_{p}\sim\sqrt{1/(I+I_{add})}$.
Because the structural inertia $I$ is prescribed, a decreasing $f_{p}$
suggests an increasing added-mass inertia, $I_{add}$. This is expected because
of the way we pitch the wings in the experiments (see the inset of figure 3).
As $\Lambda$ increases, the accelerated fluid near the wing root and the wing
tip produces more moments due to the increase of the moment arm, which
amplifies the added-mass effect. The peak added-mass moment is estimated to be
around 2%, 3%, and 5% of the peak total moment for the $\Lambda=0^{\circ}$,
$10^{\circ}$, and $20^{\circ}$ wings, respectively. Because this effect is
small compared to the structural and vortex-induced forces, we will not
quantify this added-mass effect further in the present study but will leave it
for future work.

###  3.4 Onset of flow-induced oscillations

Figure 5: Temporal evolution of (_a_) the pitching angle $\theta$, (_b_) the
fluid moment $C_{M}$, and the stiffness moment $k^{*}\theta^{*}$ near the Hopf
point for the $\Lambda=15^{\circ}$ swept wing. The vertical gray dashed line
indicates the time instant ($t=645$ s) at which $Ca$ is increased above the
Hopf point. (_c_) Static moment coefficients of unswept and swept wings.
Inset: The predicted Hopf point based on the static stall angle and the
corresponding moment, $C_{M_{s}}/\theta_{s}^{*}$, versus the measured Hopf
point, $k_{H}^{*}$. The black dashed line shows a 1:1 scaling.

In figure 3, we have observed that the Hopf point of unswept and swept wings
can be roughly divided into two groups (figure 3, G1 & G2). In this section,
we explain this phenomenon. Figure 5(_a_) and (_b_) shows the temporal
evolution of the pitching angle, $\theta(t)$, the fluid moment, $C_{M}(t)$,
and the stiffness moment, $k^{*}\theta^{*}(t)$, for the $\Lambda=15^{\circ}$
swept wing as the Cauchy number is increased past the Hopf point. We see that
the wing undergoes small amplitude oscillations around the divergence angle
just prior to the Hopf point ($t<645$ s). The divergence angle is lower than
the static stall angle, $\theta_{s}$, and so we know that the flow stays
mostly attached, and the fluid moment, $C_{M}$, is balanced by the stiffness
moment, $k^{*}\theta^{*}$ (figure 5 _b_). When the Cauchy number,
$Ca=1/k^{*}$, is increased above the Hopf point (figure 5 _a_ , $t>645$ s),
$k^{*}\theta^{*}$ is no longer able to hold the pitching angle below
$\theta_{s}$. Once the pitching angle exceeds $\theta_{s}$, stall occurs and
the wing experiences a sudden drop in $C_{M}$. The stiffness moment,
$k^{*}\theta^{*}$, loses its counterpart and starts to accelerate the wing to
pitch towards the opposite direction. This acceleration introduces
unsteadiness to the system and the small-amplitude oscillations gradually
transition to large-amplitude LCOs over the course of several cycles, until
the inertial moment kicks in to balance $k^{*}\theta^{*}$ (see also figure 4
_b_). This transition process confirms the fact that the onset of large-
amplitude LCOs depends largely on the _static_ characteristics of the wing –
the LCOs are triggered when the static stall angle is exceeded.

The triggering of flow-induced LCOs starts from $\theta$ exceeding the static
stall angle after $k^{*}$ is decreased below the Hopf point, causing $C_{M}$
to drop below $k^{*}\theta^{*}$. At this value of $k^{*}$, the slope of the
static stall point should be equal to the stiffness at the Hopf point,
$k^{*}_{H}$ (i.e. $C_{M_{s}}=k^{*}_{H}\theta^{*}$, where $C_{M_{s}}$ is the
static stall moment). This argument is verified by figure 5(_c_), in which we
replot the static moment coefficients of unswept and swept wings from figure
2(_b_) (error bars omitted for clarity), together with the corresponding
$k^{*}_{H}\theta^{*}$. We see that the $k^{*}_{H}\theta^{*}$ lines all roughly
pass through the static stall points ($\theta_{s}^{*}$, $C_{M_{s}}$) of the
corresponding $\Lambda$. Note that $k^{*}_{H}\theta^{*}$ of
$\Lambda=15^{\circ}$ and $20^{\circ}$ overlap with each other. Similar to the
trend observed for the Hopf point in figure 3, the static stall moment
$C_{M_{s}}$ can also be divided into two groups, with the unswept wing
($\Lambda=0^{\circ}$) being in G2 and all the other wings
($\Lambda=10^{\circ}$ – $25^{\circ}$) being in G1 (see also figure 2 _b_). The
inset compares the predicted Hopf point, $C_{M_{s}}/\theta_{s}^{*}$, with the
measured Hopf point, $k_{H}^{*}$, and we see that data closely follow a 1:1
relationship. This reinforces the argument that the onset of flow-induced LCOs
is shaped by the static characteristics of the wing, and that this explanation
applies to both unswept and swept wings.

It is worth noting that Negi et al. (2021) performed global linear stability
analysis on an aeroelastic wing and showed that the aeroelastic instability is
triggered by a zero-frequency linear divergence mode. This agrees in part with
our experimental observation that the flow-induced oscillations emerge from
the static divergence state. However, as we have discussed in this section,
the onset of large-amplitude aeroelastic oscillations in our system occurs
when the divergence angle exceeds the static stall angle, whereas no stall is
involved in the study of Negi et al. (2021). In fact, Negi et al. (2021)
focused on laminar separation flutter, where the pitching amplitude is small
($A\sim 6^{\circ}$). In contrast, we focus on large-amplitude
($45^{\circ}<A<120^{\circ}$) flow-induced oscillations.

###  3.5 Power coefficient map and system stability

Figure 6: (_a_ -_e_) Power coefficient maps of prescribed sinusoidal
oscillations overlaid by the bifurcation diagrams of elastically mounted
unswept and swept wings. $\triangleright$: increasing $Ca$, $\triangleleft$:
decreasing $Ca$. (_f_) Neutral power transfer curves for unswept and swept
wings. The black star represents the case $U^{*}=1.87$ ($f_{p}^{*}=0.085$),
$A=1.05$ ($60^{\circ}$), where stereo PIV measurements are taken.

In this section, we analyze the stability of elastically mounted unswept and
swept wings from the perspective of energy transfer. Menon & Mittal (2019) and
Zhu et al. (2020) have shown numerically and experimentally that the flow-
induced oscillations of elastically mounted wings can only sustain when the
net energy transfer between the ambient fluid and the elastic mount equals
zero. To map out this energy transfer for a large range of pitching
frequencies and amplitudes, we _prescribe_ the pitching motion of the wing
using a sinusoidal profile

$\theta=A\sin(2\pi f_{p}t),$ (10)

where $0\leq A\leq 2.5$ rad and $0.15~{}\mathrm{Hz}\leq f_{p}\leq
0.6~{}\mathrm{Hz}$. The fluid moment $C_{M}$ measured with these prescribed
sinusoidal motions can be directly correlated to those measured in the passive
flow-induced oscillations because the flow-induced oscillations are near-
sinusoidal (see §3.2, and figure 5 _a_ , $t>700$ s). By integrating the
governing equation of the passive system 2 over $n=20$ cycles and taking the
cycle average (Zhu et al., 2020), we can get the power coefficient of the
system

$C_{p}=\frac{f_{p}^{*}}{n}\int_{t_{0}}^{t_{0}+nT}(C_{M}\dot{\theta}^{*}-b^{*}\dot{\theta}^{*2})~{}dt^{*},$
(11)

where $t_{0}$ is the starting time, $T$ is the pitching period and
$t^{*}=tU_{\infty}/c$ is the non-dimensional time. In this equation, the
$C_{M}\dot{\theta}^{*}$ term represents the power injected into the system
from the free-stream flow, whereas the $b^{*}\dot{\theta}^{*2}$ term
represents the power dissipated by the structural damping of the elastic
mount. The power coefficient maps of unswept and swept wings are shown in
figure 6(_a_ -_e_). In these maps, orange regions correspond to $C_{p}>0$,
where the power injected by the ambient flow is higher than that dissipated by
the structural damping. On the contrary, $C_{p}<0$ in the blue regions. The
colored dashed lines indicate the $C_{p}=0$ contours, where the power
injection balances the power dissipation, and the system is in equilibrium.
The $C_{p}=0$ equilibrium boundary can be divided into three branches. Zhu et
al. (2020) have shown that for unswept wings, the top branch corresponds to a
stable LCO solution for the structural oscillation mode, the middle branch
represents an unstable LCO solution for the structural mode, but a stable LCO
solution for the hydrodynamic mode, and the bottom branch is a fixed point
solution.

To correlate the power coefficient maps of prescribed oscillations with the
stability boundaries of flow-induced oscillations, we overlay the bifurcation
diagrams of the passive system from figure 3 onto figure 6(_a_ -_e_). The
measured pitching frequencies, $f_{p}$, are used to calculate the non-
dimensional velocity, $U^{*}$, for large-amplitude LCOs (filled triangles).
Because it is difficult to measure frequencies of fixed points and small-
amplitude oscillations, we use the calculated structural frequency, $f_{s}$,
to evaluate $U^{*}$ for non-LCO data points (hollow triangles). Figure 6(_a_
-_e_) show that for all the wings tested, the flow-induced large-amplitude
LCOs match well with the top branch of the $C_{p}=0$ curve, indicating the
broad applicability of the energy approach for both unswept and swept wings,
and confirming that this instability is a structural mode, as seen in the
frequency response (figure 4 _a_). This correspondence was also observed by
Menon & Mittal (2019) and Zhu et al. (2020) and is expected for instabilities
that are well-described by sinusoidal motions (Morse & Williamson, 2009). The
small discrepancies for large sweep angles can be attributed to the low
$C_{p}$ gradient near $C_{p}=0$. The junction between the top and the middle
$C_{p}=0$ branches, which corresponds to the saddle-node point, stays
relatively sharp for $\Lambda=0^{\circ}$ – $15^{\circ}$ and becomes smoother
for $\Lambda=20^{\circ}$ – $25^{\circ}$. These smooth turnings result in a
smooth transition from the structural mode to the hydrodynamic mode, giving
rise to the structural-hydrodynamic mode discussed in §3.3.

The $C_{p}=0$ curves for $\Lambda=0^{\circ}$ – $25^{\circ}$ are summarized in
figure 6(_f_). It is seen that the trend of the top branch is similar to that
observed in figure 3 for large-amplitude LCOs. The location of the junction
between the top branch and the middle branch changes non-monotonically with
$\Lambda$, which accounts for the non-monotonic behavior of the saddle-node
point. In addition, figures 6(_a_ -_e_) show that the maximum power transfer
from the fluid also has a non-monotonic dependency on the sweep angle (see the
shade variation of the positive $C_{p}$ regions as a function of the sweep
angle), with an optimal sweep angle at $\Lambda=10^{\circ}$, which might
inspire future designs of higher efficiency oscillating-foil energy harvesting
devices.

###  3.6 Force, moment and three-dimensional flow structures

Figure 7: (_a_) Phase-averaged aerodynamic moment coefficients, $C_{M}$, and
(_b,c_) force coefficients, $C_{N}$, $C_{T}$, $C_{L}$ and $C_{D}$, measured at
$f_{p}^{*}=0.085$, $A=1.05$ ($60^{\circ}$) for the $\Lambda=0^{\circ}$,
$10^{\circ}$ and $20^{\circ}$ wings, corresponding to the black star case in
figure 6(_f_). (_d-f_) Phase-averaged moment coefficients, $C_{M}$, and power
coefficients, $C_{P}$, for $\Lambda=0^{\circ}$, $10^{\circ}$ and $20^{\circ}$.
Green panels represent positive power input regions, where $C_{P}>0$. Gray
dashed lines and dotted lines represent the normalized pitching angle,
$\theta/A$, and pitching velocity, $\dot{\theta}/(2\pi f_{p}A)$, respectively.
Note that $t/T=0$ corresponds to $\theta=0$ (see the gray dashed curve).

In the previous section, §3.5, we have established the connection between
prescribed oscillations and flow-induced instabilities using the energy
approach. However, the question remains what causes the differences in the
power coefficients measured for prescribed pitching wings with different sweep
angles (figure 6). In this section, we analyze the aerodynamic force, moment
and the corresponding three-dimensional flow structures to gain more insights.
We focus on one pitching case, $A=1.05$ ($60^{\circ}$) and $f^{*}_{p}=0.085$
(i.e. the black star on figure 6 _f_), and three sweep angles,
$\Lambda=0^{\circ}$, $10^{\circ}$ and $20^{\circ}$. This particular pitching
kinematic is selected because it sits right on the $C_{p}=0$ curve for
$\Lambda=0^{\circ}$ but in the positive $C_{p}$ region for
$\Lambda=10^{\circ}$ and in the negative $C_{p}$ region for
$\Lambda=20^{\circ}$ (see figure 6 _a,b,d,f_).

Phase-averaged coefficients of the aerodynamic moment, $C_{M}$, the normal
force, $C_{N}$, the tangential force, $C_{T}$, the lift force, $C_{L}$, and
the drag force, $C_{D}$, are plotted in figure 7(_a-c_), respectively. Similar
to the three-dimensional velocity fields, the moment and force measurements
are phase-averaged over 25 cycles. We see that the moment coefficient (figure
7 _a_) behaves differently for different sweep angles, whereas the shape of
other force coefficients (figure 7 _b,c_) does not change with sweep angle,
resembling the trend observed in the static measurements (figure 2). The
observation that the wing sweep ($\Lambda=0^{\circ}$ to $25^{\circ}$) has
minimal effects on the aerodynamic force generation is non-intuitive, as one
would assume that the sweep-induced spanwise flow can enhance spanwise
vorticity transport in the leading-edge vortex and thereby alter the LEV
stability as well as the resultant aerodynamic load. However, our measurements
show the opposite, a result which is backed up by the experiments of heaving
(plunging) swept wings by Beem et al. (2012) ($\Lambda=0^{\circ}$ to
$45^{\circ}$) and Wong et al. (2013) ($\Lambda=0^{\circ}$ and $\pm
45^{\circ}$), simulations of pitching swept wings by Visbal & Garmann (2019)
($\Lambda=0^{\circ}$ to $30^{\circ}$), and simulations of fin-like pitch-heave
swept wings by Zurman-Nasution et al. (2021) ($\Lambda=0^{\circ}$ to
$40^{\circ}$), where the spanwise flow has been shown to exist but to have no
effect on the aerodynamic force. We also analyze aerodynamic forces for
different sweep angles and other wing kinematics and observe similar results
(not shown in this manuscript). The collapse of the normal force, $C_{N}$, at
different sweep angles suggests that the wing sweep regulates the aerodynamic
moment, $C_{M}$, by changing the moment arm, $d_{M}$, as $C_{M}=C_{N}d_{M}$.
This argument will be revisited later when we discuss the leading-edge vortex
and tip vortex dynamics.

Figure 7(_a_) shows that as the sweep angle increases, the moment coefficient,
$C_{M}$, peaks at a later time in the cycle, and has an increased maximum
value. To further analyze $C_{M}$ and its effects on the power coefficient,
$C_{P}$, for different wings sweeps, we compare $C_{M}$ and $C_{P}$ for
$\Lambda=0^{\circ}$, $10^{\circ}$ and $20^{\circ}$ in figure 7(_d-f_),
respectively. Note that here we define the power coefficient as
$C_{P}=C_{M}\dot{\theta}^{*}$, which is different from equation 11 in a way
that this $C_{P}$ is time-dependent instead of cycle-averaged, and that the
power dissipated by the structure, $b^{*}\dot{\theta}^{*2}$ is not considered
(this power dissipation is small because a small $b^{*}$ is used in the
experiments). The normalized pitching angle, $\theta/A$, and pitching
velocity, $\dot{\theta}/(2\pi f_{p}A)$, are also plotted for reference. We see
that at the beginning of the cycle ($0\leq t/T<0.15$), $C_{M}(t/T)$ grows
near-linearly for all three wings. Because $\dot{\theta}>0$ for the first
quarter cycle, the $x$-intercept of $C_{M}$ determines the starting point of
the positive $C_{P}(t/T)$ region, corresponding to the left edge of the green
panels in the figures. The $C_{P}>0$ region starts at $t/T=0$ for the unswept
wing as $C_{M}$ has a near-zero $y$-intercept. For the $\Lambda=10^{\circ}$
swept wing, because $C_{M}$ has a small positive $y$-intercept, the $C_{P}>0$
region starts even before $t/T=0$. On the contrary, the $C_{P}>0$ region
starts after $t/T=0$ for the $\Lambda=20^{\circ}$ swept wing due to a small
negative $y$-intercept of $C_{M}$. Owing to the combined effect of an
increasing $C_{M}$ and a decreasing $\dot{\theta}$, the power coefficient
peaks around $t/T=0.125$ for all the wings. The maximum $C_{P}$ of the
$\Lambda=10^{\circ}$ wing is slightly higher than that of the other two wings,
due to a slightly higher $C_{M}$.

As the pitching cycle continues, $C_{M}(t/T)$ peaks around $t/T=0.15$, 0.17
and 0.28 for $\Lambda=0^{\circ}$, $10^{\circ}$ and $20^{\circ}$, respectively.
The pitch reversal occurs at $t/T=0.25$, where $\theta$ reaches its maximum
and $\dot{\theta}$ switches its sign to negative. Because the pitching
velocity is now negative, the green panels terminate as $C_{P}$ drops below
zero, suggesting that $C_{M}$ starts to dissipate energy into the ambient
fluid. However, because $C_{M}$ continues to grow after $t/T=0.25$ for the
$\Lambda=20^{\circ}$ wing, it generates a much more negative $C_{P}$ as
compared to the wings with a lower sweep angle. Figure 7(_a_) shows that
$C_{M}$ decreases faster for the $\Lambda=10^{\circ}$ wing than the unswept
wing at $0.25\leq t/T<0.5$. This difference results in a less negative $C_{P}$
for the $\Lambda=10^{\circ}$ wing as compared to the $\Lambda=0^{\circ}$ wing.
The faster decrease of $C_{M}$ for the $\Lambda=10^{\circ}$ wing also makes it
the first to switch back to positive power generation, where $C_{M}$ and
$\dot{\theta}$ are both negative. The same story repeats after $t/T=0.5$ due
to the symmetry of the pitching cycle. In summary, we see that subtle
differences in the alignment of $C_{M}$ and $\dot{\theta}$ can result in
considerable changes of $C_{P}$ for different sweep angles. The start of the
$C_{P}>0$ region is determined by the phase of $C_{M}$, whereas the
termination of the $C_{P}>0$ region depends on $\dot{\theta}$. A non-monotonic
duration of the $C_{P}>0$ region (i.e. the size of the green panels) is
observed as the sweep angle increases. The cycle-averaged power coefficient,
which dictates the stability of aeroelastic systems (see §3.5), is regulated
by both the amplitude and phase of the aerodynamic moment.

Figure 8: (_a_) Moment coefficients replotted from figure 7(_a_) for half
pitching cycle. Three representative time instants $t_{1}/T=0.14$,
$t_{2}/T=0.22$ and $t_{3}/T=0.30$ are selected for studying the evolution of
the leading-edge vortex (LEV) and tip vortex (TV). (_b-d_) Phase-averaged
three-dimensional flow structures for the $\Lambda=0^{\circ}$ unswept wing,
and the $\Lambda=10^{\circ}$ and $\Lambda=20^{\circ}$ swept wings. The flow
structures are visualized with iso-$Q$ surfaces ($Q=50~{}\mathrm{s}^{-2}$) and
colored by the non-dimensional spanwise vorticity, $\omega_{z}c/U_{\infty}$.
All the flow fields are rotated by the pitching angle to keep the wing at a
zero angle of attack for better visualization of the flow structures. A video
capturing the three-dimensional flow structures for the entire pitching cycle
can be found in the supplementary material. (_e-g_) Side views and front views
of the corresponding three-dimensional LEV and TV geometries. Solid curves
represent LEVs and dotted lines represent TVs.

Next, we analyze the effect of wing sweep on the leading-edge vortex and tip
vortex dynamics and the resultant impact on the aerodynamic moment. Figure 8
shows (_a_) the moment measurements, (_b-d_) the phase-averaged three-
dimensional flow structures at $t_{1}/T=0.14$, $t_{2}/T=0.22$ and
$t_{3}/T=0.30$, and (_e-g_) the corresponding leading-edge vortex and tip
vortex geometries for the $\Lambda=0^{\circ}$, $10^{\circ}$ and $20^{\circ}$
wings. The three equally spaced time instants $t_{1}/T=0.14$, $t_{2}/T=0.22$
and $t_{3}/T=0.30$ are selected because they correspond to the times of the
formation, growth and shedding of the leading-edge vortex. The three-
dimensional flow structures are visualized using iso-$Q$ surfaces with a value
of $50~{}\mathrm{s}^{-2}$ and colored by the non-dimensional spanwise
vorticity, $\omega_{z}c/U_{\infty}$. In this view, the leading edge of the
wing is pitching towards us, but for clarity, the flow field is always plotted
with the coordinate system oriented so that the chord line is aligned with the
$x-$axis.

The initial linear growth of the moment coefficient before $t_{1}/T$ for all
three wings corresponds to the formation of a strong leading-edge vortex, as
depicted in figure 8(_b-d_) at $t_{1}/T=0.14$, which brings the lift and
moment coefficients above the static stall limit. At this stage, we see that
the structure of the leading-edge vortex is similar across different wing
sweeps, despite some minor variations near the wing tip. For the unswept wing,
the LEV stays mostly attached along the wing span, whereas for the two swept
wings, the LEV starts to detach near the tip region (see the small holes on
the feeding shear layer near the wing tip). A positive vortex tube on the
surface near the trailing edge is observed for all three wings, along with the
negative vortex tubes shed from the trailing edge. We also observe a
streamwise-oriented tip vortex wrapping over the wing tip, and this tip vortex
grows stronger with the sweep angle, presumably due to the higher tip velocity
associated with the larger wing sweep. Another possible cause for a stronger
TV at a higher sweep angle is that the effective angle of attack becomes
higher at the wing tip as the wing sweep increases.

The tracking of the vortex geometry (figure 8 _e-g_) provides a more
quantitative measure to analyze the LEV and TV dynamics. We see that at
$t_{1}/T=0.14$, the LEVs for all three wings are mostly aligned with the
leading edge except for the tip region ($z/c=0$). For the two swept wings, the
LEV also stays closer to the leading edge near the wing root ($z/c=3$). Due to
the high wing sweep of the $\Lambda=20^{\circ}$ wing, a small portion of the
LEV falls behind the pivot axis, presumably contributing to a negative moment.
However, the mean distance between the LEV and the pivot axis (i.e. the LEV
moment arm) stays roughly constant across different wing sweeps, potentially
explaining the agreement between the $C_{M}$ for different wings during the
linear growth region. On the other hand, the tip vortex moves downstream as
the wing sweep increases due to the wing geometry. For the unswept wing and
the $\Lambda=10^{\circ}$ swept wing, the majority of the tip vortex stays
behind the pivot axis. For the $\Lambda=20^{\circ}$ swept wing, the TV stays
entirely behind the pivot axis. As a result, the TV mostly contributes to the
generation of negative moments, which counteracts the LEV moment contribution.

At $t_{2}/T=0.22$, figure 8(_b_) and the front view of figure 8(_e_) show that
the LEV mostly detaches from the wing surface for the unswept wing except for
a small portion near the wing tip, which stays attached. A similar flow
structure was observed by Yilmaz & Rockwell (2012) for finite-span wings
undergoing linear pitch-up motions, and by Son et al. (2022a) for high-aspect-
ratio plunging wings. For the $\Lambda=10^{\circ}$ wing, this small portion of
the attached LEV shrinks (see the front view of figure 8 _f_). The top portion
of the LEV near the wing root is also observed to stay attached to the wing
surface as compared to the $\Lambda=0^{\circ}$ case. For the
$\Lambda=20^{\circ}$ wing, as shown by the front view of figure 8(_g_), the
attached portion of the LEV near the wing tip further shrinks and almost
detaches, while the top portion of the LEV also attaches to the wing surface,
similar to that observed for $\Lambda=10^{\circ}$. The shrinking of the LEV
attached region near the wing tip as a function of the wing sweep is
presumably caused by the decreased anchoring effect of the tip vortex. The
shrinking of the attached LEV could also be a result of an increased effective
angle of attack. The side views of figure 8(_e-g_) show that the LEV moves
towards the pivot axis at this time instant. The swept wing LEVs have slightly
longer mean moment arms due to their attached portions near the wing root.
This is more prominent for the $\Lambda=20^{\circ}$ wing, potentially
explaining the $C_{M}$ of $\Lambda=20^{\circ}$ exceeding the other two wings
at $t_{2}/T$. The tip vortex moves upwards and outwards with respect to the
wing surface from $t_{1}/T$ to $t_{2}/T$.

During the pitch reversal ($t_{3}/T=0.30$), the LEV further detaches from the
wing surface, and the TV also starts to detach. For the unswept wing, the LEV
mostly aligns with the pivot axis except for the tip portion, which still
remains attached. For the $\Lambda=10^{\circ}$ swept wing, the LEV also
roughly aligns with the pivot axis, with both the root and the tip portions
staying near the wing surface, forming a more prominent arch-like shape (see
the front view of figure 8 _f_) as compared to the previous time step. For the
$\Lambda=20^{\circ}$ wing, the root portion of the LEV stays attached and
remains far in front of the pivot axis. The LEV detaches near the wing tip and
joins with the detached TV, as shown by figure 8(_d_) and the front and top
views of figure 8(_g_). The attachment of the LEV near the wing root and the
detachment of the TV near the wing tip both contribute to a more positive
$C_{M}$, as compared to the other two wings with lower sweep. The change of
the LEV geometry as a function of the sweep angle can be associated with the
arch vortices reported by Visbal & Garmann (2019). In their numerical study,
it has been shown that for pitching unswept wings with free tips on both ends,
an arch-type vortical structure began to form as the pitch reversal started
(see their figure 6 _c_). In our experiments, the wings have a free tip and an
endplate (i.e. a wing-body junction, or symmetry plane). Therefore, the
vortical structure shown in figure 8(_b_) is equivalent to one-half of the
arch vortex. If we mirror the flow structures about the wing root (i.e. the
endplate), we can get a complete arch vortex similar to that observed by
Visbal & Garmann (2019). For swept wings, we observe one complete arch vortex
for both $\Lambda=10^{\circ}$ (figure 8 _c_) and $20^{\circ}$ (figure 8 _d_).
Again, if we mirror the flow structures about the wing root, there will be two
arch vortices for each swept wing, agreeing well with the observation of
Visbal & Garmann (2019) (see their figures 10 _c_ and 13 _c_). Moreover,
Visbal & Garmann (2019) reported that for swept wings, as $\Lambda$ increases,
the vortex arch moves towards the wing tip, which is also seen in our
experiments (compare the front views of figure 8 _e-g_).

###  3.7 Insights obtained from moment partitioning

We have shown in the previous section, §3.6, that the aerodynamic moment is
jointly determined by the leading-edge vortex and the tip vortex dynamics.
Specifically, the spatial locations and geometries of the LEV and TV, as well
as the vortex strength, have a combined effect on the unsteady aerodynamic
moment. To obtain further insights into this complex combined effect, we use
the Force and Moment Partitioning Method (FMPM) to analyze the three-
dimensional flow fields.

Figure 9: Iso-surface plots of three-dimensional influence potentials for
(_a_) the $\Lambda=0^{\circ}$ unswept wing, (_b_) the $\Lambda=10^{\circ}$
swept wing, and (_c_) the $\Lambda=20^{\circ}$ swept wing. (_d-f_) The
corresponding side views, with the wing boundaries outlined by yellow dotted
lines and the pitching axes indicated by green dashed lines.

As we discussed in §2.4, the first step of applying FMPM is to construct an
‘influence potential’, $\phi$. We solve equation 6 numerically using the
MATLAB Partial Differential Equation Toolbox (Finite Element Method, code
publicly available on MATLAB File Exchange). We use a 3D domain of $10c\times
10c\times 20c$, and a mesh resolution of $0.02c$ on the surface of the wing
and $0.1c$ on the outer domain. We visualize the calculated three-dimensional
influence field, $\phi$, for the $\Lambda=0^{\circ}$, $10^{\circ}$ and
$20^{\circ}$ wings using iso-$\phi$ surfaces in figure 9(_a-c_). Figure 9(_d-
f_) illustrates the corresponding side views, with the wing boundaries
outlined by yellow dotted lines and the pitching axes indicated by green
dashed lines. We see that for the unswept wing, the iso-$\phi$ surfaces show
symmetry with respect to the pivot axis and the wing chord, resulting in a
quadrant distribution of the influence field. The magnitude of $\phi$ peaks on
the wing surface and decreases towards the far field. The slight asymmetry of
$\phi$ with respect to the pitching axis (see figure 9 _d_) is caused by the
difference between the rounded leading edge and the sharp trailing edge of the
NACA 0012 wing (see also the 2D influence field reported in Zhu et al.
(2023)). The size of the iso-$\phi$ surfaces stays relatively constant along
the wing span, except at the wing tip, where the surfaces wrap around and seal
the tube.

As the sweep angle is increased to $\Lambda=10^{\circ}$ and $20^{\circ}$, we
see that the quadrant distribution of the influence field persists. However,
the iso-$\phi$ surfaces form funnel-like shapes on the fore wing and teardrop
shapes on the aft wing. This is caused by the variation of the effective pivot
axis along the wing span. Figure 9(_e_) and (_f_) show that, for swept wings,
the negative $\phi$ regions extend over the entire chord near the wing root,
even behind the pitching axis. Similarly, the positive $\phi$ regions (almost)
cover the entire wing tip and even spill over in front of the pitching axis.
As we will show next, this behavior of the $\phi$ field for swept wings will
result in some non-intuitive distributions of the aerodynamic moment. In
addition, the magnitude of the $\phi$ field is observed to increase with the
sweep angle, due to the increase of the effective moment arm (Zhu et al.,
2021).

Figure 10: (_a-c_) Phase-averaged iso-$Q$ surfaces ($Q=50~{}\mathrm{s}^{-2}$)
for the $\Lambda=0^{\circ}$ unswept wing and the $\Lambda=10^{\circ}$ and
$20^{\circ}$ swept wings, colored by the vorticity-induced moment density,
$-2Q\phi$ ($\mathrm{m^{2}~{}s^{-2}}$), at $t_{1}/T=0.14$, $t_{2}/T=0.22$ and
$t_{3}/T=0.30$. Note that the wings and flow fields are rotated in the
spanwise direction to maintain a zero angle of attack, for a better view of
the flow structures. (_d-f_) Spanwise distributions of the vorticity-induced
moment for the three wings at the three representative time instants, obtained
by integrating $-2Q\phi$ at different spanwise locations.

We multiply the three-dimensional $Q$ field by the influence field, $\phi$,
and get the spanwise moment (density) distribution field, $-2Q\phi$. To
visualize the moment distributions, we recolor the same iso-$Q$ surface plots
shown in figure 8 with the moment density, $-2Q\phi$, which are shown in
figure 10(_a-c_). As before, the wings and flow fields are rotated by $\theta$
so that we are always looking from a viewpoint normal to the chord line,
giving a better view of the flow structures. In these iso-$Q$ surface plots,
red regions indicate that the vortical structure induces a positive spanwise
moment, whereas blue regions represent the generation of a negative spanwise
moment. In between red and blue regions, white regions have zero contribution
to the spanwise moment.

At $t_{1}/T=0.14$ (figure 10 _a_), as expected, we see that the entire LEV on
the unswept wing is generating a positive moment. For the $\Lambda=10^{\circ}$
swept wing, however, the LEV generates a near-zero moment near the wing tip,
and for the $\Lambda=20^{\circ}$ swept wing, the tip region of the LEV
contributes a negative moment due to the non-conventional distribution of the
$\phi$ field. The TV generates almost no moment for the unswept wing, but
contributes a negative moment for the swept wings. The vortex tube formed near
the trailing edge of the wing surface contributes entirely to negative moments
for the unswept wing, but its top portion starts to generate positive moments
as the sweep angle increases. The contributions of each vortical structure on
the moment generation for the three wings become more clear if we plot the
spanwise distribution of the vorticity-induced moment.

By integrating the moment distribution field $-2Q\phi$ over the horizontal
($x,y$)-plane at each spanwise location, $z$, we are able to obtain the
spanwise distribution of the vorticity-induced moment, shown in figure 10(_d-
f_). For the unswept wing, $\Lambda=0^{\circ}$, figure 10(_d_) shows that the
LEV generates a near-uniform positive moment across the span. As the sweep
angle increases ($\Lambda=10^{\circ}$), the LEV generates a higher positive
moment near the wing root, and the TV starts to generate a negative moment.
For the $\Lambda=20^{\circ}$ wing, this trend persists. It is also interesting
to see that the spanwise moment distribution curves for the three wings
intersect around the mid span, where the effective pivot axis coincides at the
mid chord. For the two swept wings, the more positive moments near the wing
root counteract the negative LEV and TV contributions near the wing tip,
resulting in a similar overall moment as compared to the unswept wing. The
FMPM thus quantitatively explains why the three wings generate similar
unsteady moments at this time instant (figure 8 _a_).

At $t_{2}/T=0.22$ (figure 10 _b_), the LEV starts to detach and moves towards
the pitching axis. As discussed in the previous section, §3.6, the LEV forms a
half-arch for the unswept wing, with only the tip region staying attached, and
a complete arch for swept wings, with both the root and tip regions staying
attached. These arch-like LEV geometries, together with the special shapes of
the three-dimensional influence field, lead to some special distributions of
the aerodynamic moments. For the unswept wing, the color of the LEV becomes
lighter as compared to the $t_{1}/T$ case, indicating a decreasing
contribution to positive moments. However, the attached portion of the LEV
still generates a positive moment as it remains attached, close to the wing,
and in front of the pitching axis. Comparing the two swept wing cases, the LEV
for the $\Lambda=20^{\circ}$ wing generates more positive moments near the
wing root as compared to the $\Lambda=10^{\circ}$ wing due to the magnitude of
the $\phi$ field (figure 9). The TVs for the three wings behave similarly to
the cases at $t_{1}/T$. The aft wing vortex tube on the wing surface breaks
into two smaller tubes. Because of their small volumes, we do not expect them
to affect the total moment generation. Figure 10(_e_) shows that the large
part of the LEV does not contribute to any moment generation for the unswept
wing – only the tip region ($0\leq z/c\leq 1$) generates positive moments. As
compared to $t_{1}/T$, the LEV generates more positive moments near the wing
root for the two swept wings, especially for the $\Lambda=20^{\circ}$ wing,
and the TV generates slightly more negative moments. The overall trend
observed in figure 10(_e_) further explains the moment measurements shown in
figure 8(_a_), where the $\Lambda=20^{\circ}$ wing produces the highest
$C_{M}$, followed by the $\Lambda=10^{\circ}$ wing and then the unswept wing
at $t_{2}/T$.

At $t_{3}/T=0.30$ (figure 10 _c_), the LEV further detaches from the wing
surface. For the unswept wing, the LEV color becomes even lighter. Comparing
the temporal evolution of the LEV color for the unswept wing, we see that the
LEV progressively generates lower positive moments, agreeing well with the
decreasing moment measurement shown in figure 8(_a_). The LEV continues to
generate positive moments near the root region and negative moments near the
tip region for the $\Lambda=10^{\circ}$ swept wing, although it is largely
aligned with the pivot axis (see also the side view of figure 8 _f_). This is
again a result of the non-conventional funnel-shaped $\phi$ field near the
wing root and the teardrop-like $\phi$ field near the wing tip (figure 9 _b_
and _e_). This trend persists for the $\Lambda=20^{\circ}$ wing. However, the
LEV generates more positive moments due to its shorter distance from the
leading edge and the wing surface near the wing root. Moreover, the size of
the LEV iso-$Q$ surface also becomes larger for the $\Lambda=20^{\circ}$ wing
as compared to the previous time steps, indicating a stronger LEV and thus a
higher aerodynamic moment, which explains why the $C_{M}$ of
$\Lambda=20^{\circ}$ peaks around $t_{3}/T$ in figure 8(_a_). This is also
reflected in the spanwise moment plot in figure 10(_f_), where the LEV
generates more positive moments for the $\Lambda=20^{\circ}$ wing than the
$\Lambda=10^{\circ}$ wing. The tip vortex again behaves similarly to the
previous time steps for all three wings, although it becomes less coherent and
detaches from the wing surface.

It is worth noting that the integral of $-2Q\phi$ over the ($x,y$)-plane (i.e.
figure 10 _d-f_) also includes contributions from other vortical structures.
In figure 10(_a-c_), we can see that there are four main structures on each
wing: the LEV, the TV, the TEV, and the vortex tube on the aft wing surface.
Figure 9 shows that the amplitude of the influence field, $\phi$, is zero near
the trailing edge due to symmetry. This means that the contribution to the
moment by the TEV is negligible, because $-2Q\phi$ approaches zero in this
region and makes no contribution to the integrand. The aft wing vortex tube is
small in size compared to the LEV and TV. In addition, it is not as coherent,
because it breaks down at $t_{2}/T=0.22$. Therefore, we would expect its
contribution to the integral to be small as well.

In summary, the Force and Moment Partitioning Method enables us to associate
the complex three-dimensional vortex dynamics with the corresponding
vorticity-induced moments, and quantitatively explains the mechanisms behind
the observed differences in the unsteady moment generation, which further
drives the pitching motion of these swept wings. These insightful analyses
would not have been possible without the FMPM.

##  4 Conclusion

In this experimental study, we have explored the nonlinear flow-induced
oscillations and three-dimensional vortex dynamics of cyber-physically mounted
pitching unswept and swept wings, with the pitching axis passes through the
mid-chord point at the mid-span plane, and with the sweep angle varied from
$0^{\circ}$ to $25^{\circ}$. At a constant flow speed, a prescribed high
inertia and a small structural damping, we adjusted the wing stiffness to
systematically study the onset and extinction of large-amplitude flow-induced
oscillations. For the current selections of the pitching axis location and the
range of the sweep angle, the amplitude response revealed subcritical Hopf
bifurcations for all the unswept and swept wings, with a clustering behavior
for the Hopf point and a non-monotonic saddle-node point as a function of the
sweep angle. The flow-induced oscillations have been correlated with the
structural oscillation mode, where the oscillations are dominated by the
inertial behavior of the wing. For swept wings with high sweep angles, a
hybrid oscillation mode, namely the structural-hydrodynamic mode, has been
observed and characterized, in which the oscillations were regulated by both
the inertial moment and the fluid moment. The onset of flow-induced
oscillations (i.e. the Hopf point) has been shown to depend on the static
characteristics of the wing. The non-monotonic trend of the saddle-node point
against the sweep angle can be attributed to the non-monotonic power transfer
between the ambient fluid and the elastic mount, which further depends on the
amplitude and phase of the unsteady aerodynamic moment. Force and moment
measurements have shown that, perhaps surprisingly, the wing sweep has a
minimal effect on the aerodynamic forces and it was therefore inferred that
the wing sweep modulates the aerodynamic moment by affecting the moment arm.
Phase-averaged three-dimensional flow structures measured using stereoscopic
PIV have been analyzed to characterize the dynamics of the leading-edge vortex
and tip vortex. Finally, by employing the Force and Moment Partitioning Method
(FMPM), we have successfully correlated the complex LEV and TV dynamics with
the resultant aerodynamic moment in a quantitative manner.

In addition to reporting new observations and providing physical insights on
the effects of moderate wing sweep in large-amplitude aeroelastic
oscillations, the present study can serve as a source of validation data for
future theoretical/computational models. Furthermore, the optimal sweep angle
($\Lambda=10^{\circ}$) observed for promoting flow-induced oscillations may
have engineering implications. For example, one should avoid this sweep angle
for aero-structure designs to stay away from aeroelastic instabilities. On the
other hand, this angle could potentially be employed for developing higher-
efficiency flapping-foil energy-harvesting devices. Lastly, the use of FMPM to
analyze (especially three-dimensional) flow fields obtained from PIV
experiments has shown great utility, and the results further demonstrated the
powerful capability of this emerging method to provide valuable physical
insights into vortex-dominated flows, paving the way for more applications of
this method to data from future experimental and numerical studies.

## Acknowledgments

This work is funded by the Air Force Office of Scientific Research, Grant
FA9550-21-1-0462, managed by Dr. Gregg Abate. We acknowledge the helpful
discussions with Rajat Mittal, Karthik Menon, and Sushrut Kumar.

## Declaration of interests

The authors report no conflict of interest.

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