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

Abstract We study experimentally the aeroelastic instability boundaries and three-dimensional vortex dynamics of pitching swept wings, with the sweep angle ranging from 0$^\circ$ to 25$^\circ$. 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 analysed to explain the differences in power extraction for different swept wings. Finally, we employ a physics-based force and moment partitioning method to correlate quantitatively the three-dimensional vortex dynamics with the resultant unsteady aerodynamic moment.


Introduction
The fluid-structure interaction (FSI) of elastically mounted pitching wings can lead to largeamplitude 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 flappingwing 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 threedimensional 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.

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  ×  ×  = 0.8 m × 0.6 m × 4.0 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, , damping, , and inertia, ) 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, , 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 () and the damping moment ( ) obtained from the previous time step to get the total moment.Next, we divide this total moment by the desired inertia () to get the acceleration ( ) at the present time step.This acceleration is then integrated once to get the velocity ( ) and twice to get the pitching angle ().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  = 0.3 m and a chord length of  = 0.1 m, which results in a physical aspect ratio of  = 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 1.5.The chord-based Reynolds number is defined as  ≡  ∞ /, where  ∞ is the free-stream velocity,  and  are water density and dynamic viscosity, respectively.We set the free-stream velocity to be  ∞ = 0.5 m s −1 for all the experiments (except for particle image velocimetry measurements, see §2.2), which results in a constant Reynolds number of  = 50 000, matching the  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 / = 0.5 of the mid-span plane / = 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, Λ, is defined as the angle between the leading edge and the vertical axis.Five wings with Λ = 0 • (unswept wing), 10 • , 15 • , 20 • and 25 • (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 Λ = 25 • 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.

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  ∞ = 0.3 m s −1 to enable higher temporal measurement resolution.The chord-based Reynolds number is consequently decreased to  = 30 000.It has been shown by Zhu et al. (2020, see their appendix) that the variation of  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 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 × 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 × 64 pixels, two passes at 32 × 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 -axis (King et al. 2018;Zhong et al. 2021a).Due to the limitation of the traversing range, three measurement volumes (figure 1b, 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 (∼ 3 × 3 × 3.5).The velocity fields of three wing models (Λ = 0 • , 10 • and 20 • ) are measured.For the two swept wings (Λ = 10 • and 20 • ), the laser volumes are offset horizontally to compensate for the sweep angle (see the bottom subfigure of figure 1b).

Governing equations and non-dimensional parameters
The one-degree-of-freedom aeroelastic system considered in the present study has a governing equation   +   +  = , (2.1) where , , and  are the angular position, velocity and acceleration, respectively. =   +   is the effective inertia, where   is the physical inertia of the wing and   is the virtual inertia that we prescribe with the CPS.Because the friction is negligible in our system, the effective structural damping, , equals the virtual damping   in the CPS. is the effective torsional stiffness and it equals the virtual stiffness   .Equation 2.1 resembles a forced torsional spring-mass-damper system, where the fluid moment, , 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: where We should note that the inverse of the non-dimensional stiffness is equivalent to the Cauchy number,  = 1/ * , and the non-dimensional inertia,  * , is analogous to the mass ratio between the wing and the surrounding fluid.We define the non-dimensional velocity as  * =  ∞ /(2   ), where   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,   and   , respectively.The resultant lift and drag forces are (2.4) We further normalize the normal force, tangential force, lift and drag to get the corresponding force coefficients where n is the unit vector normal to the boundary,  −   is the location vector pointing from the pitching axis   towards location  on the airfoil surface, and 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 2.6 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  field (Hunt et al. 1988;Jeong & Hussain 1995) where  is the second invariant of the velocity gradient tensor,  is the vorticity tensor and S is the strain-rate tensor.The vorticity-induced moment can be evaluated by where ∫  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, −2 (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.

Static characteristics of unswept and swept wings
The static lift and moment coefficient,   and   , are measured for the unswept (Λ = 0 • ) and swept wings (Λ = 10 • -25 • ) at  = 50 000 and the results are shown in figure 2. In figure 2(a), we see that the static lift coefficient,   (), has the same behavior for all sweep angles, despite some minor variations for angles of attack higher than the static stall angle   = 12 • (0.21 rad).The collapse of   () across different swept wings agrees with the classic 'independence principle' (Jones 1947) (i.e.  ∼ cos 2 Λ) at relatively small sweep angles.Figure 2(b) shows that, for any fixed angle of attack, the static moment coefficient,   , increases with the sweep angle, Λ.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   for  = 0.14 -0.26.It is seen that the   curves cluster into two groups, with the unswept wing (Λ = 0 • ) being in Group 2 (G2) and all the other swept wings (Λ = 10 • -25 • ) 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   and   , presumably due to free-stream turbulence in the water tunnel.

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  * .For these reasons, we choose to focus on one high-inertia case ( * = 10.6) in the present study.In the experiments, the free-stream velocity is maintained at  ∞ = 0.5 m s −1 .We fix the structural damping of the system at a small value,  * = 0.13, keep the initial angle of attack at zero, and use the Cauchy number, , as the control parameter.To test for the onset of LCOs, we begin the test with a high-stiffness virtual spring (i.e.low ) and incrementally increase  by decreasing the torsional stiffness,  * .We then reverse the operation to test for the extinction of LCOs and to check for any hysteresis.The amplitude response of the system, , is measured as the peak absolute pitching angle (averaged over many pitching cycles).By this definition,  is half of the peak-to-peak amplitude.The divergence angle, , 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 shows the pitching amplitude response and the static divergence angle for swept wings with Λ = 10 • to 25 • .Data for the unswept wing (Λ = 0 • ) 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  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 , 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  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  (regime 3 ).We then decrease  and find that the large-amplitude LCOs persist even when  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 saddlenode 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 (Λ = 0 • ) being in G2 and all the other swept wings (Λ = 10 • -25 • ) 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  at which the saddle-node point occurs first extends towards a lower value (Λ = 0 • → 10 • ) but then moves back towards a higher  (Λ = 10 • → 25 • ).This indicates that increasing the sweep angle first destabilizes the system from Λ = 0 • to 10 • and then re-stabilizes it from Λ = 10 • to 25 • .This non-monotonic behavior of the saddle-node point will be revisited from a perspective of energy in §3.5.The pitching amplitude response, , follows a similar non-monotonic trend.Between Λ = 0 • and 10 • ,  is slightly higher at higher  values for the Λ = 10 • wing, whereas between Λ = 10 • and 25 • ,  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, Λ = 10 • , which promotes flow-induced oscillations of pitching swept wings.

Frequency response of the system
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 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,  *  , locks onto the calculated structural frequency,  *  , 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,  *  * , the structural damping moment,  *  * , the stiffness moment,  *  * , and the fluid moment,   .As an example, for the Λ = 10 • wing pitching at  *  = 0.069 (i.e. the filled orange triangle in figure 4a), 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,  *  , stays relatively constant for a varying .In figure 4(a), we see that for the Λ = 20 • and 25 • wings,  *  flattens near the saddle-node boundary.This flattening trend shows an emerging fluiddominated time scale, resembling a hydrodynamic mode despite the high wing inertia.Taking Λ = 20 • ,  *  = 0.068 (i.e. the filled green diamond in figure 4a) 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 lockin) 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 4a, green and dark green curves) at high  *  , 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 8a).The structural-hydrodynamic mode emerges as the fluid forcing term starts to dominate in the nonlinear oscillator.
For a fixed structural frequency,  *  , as the sweep angle increases, the measured pitching frequency,  *  , 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   ∼ √︁ 1/( +   ).Because the structural inertia  is prescribed, a decreasing   suggests an increasing addedmass inertia,   .This is expected because of the way we pitch the wings in the experiments (see the inset of figure 3).As Λ 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 Λ = 0 • , 10 • , and 20 • 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.

Onset of flow-induced oscillations
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,  (), the fluid moment,   (), and the stiffness moment,  *  * (), for the Λ = 15 • 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 ( < 645 s).The divergence angle is lower than the static stall angle,   , and so we know that the flow stays mostly attached, and the fluid moment,   , is balanced by the stiffness moment,  *  * (figure 5b).When the Cauchy number,  = 1/ * , is increased above the Hopf point (figure 5a,  > 645 s),  *  * is no longer able to hold the pitching angle below   .Once the pitching angle exceeds   , stall occurs and the wing experiences a sudden drop in   .The stiffness moment,  *  * , 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  *  * (see also figure 4b).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  exceeding the static stall angle after  * is decreased below the Hopf point, causing   to drop below  *  * .At this value of  * , the slope of the static stall point should be equal to the stiffness at the Hopf point,  *  (i.e.   =  *   * , where    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  *   * .We see that the  *   * lines all roughly pass through the static stall points ( *  ,    ) of the corresponding Λ.Note that  *   * of Λ = 15 • and 20 • overlap with each other.Similar to the trend observed for the Hopf point in figure 3, the static stall moment    can also be divided into two groups, with the unswept wing (Λ = 0 • ) being in G2 and all the other wings (Λ = 10 • -25 • ) being in G1 (see also figure 2b).The inset compares the predicted Hopf point,    / *  , with the measured Hopf point,  *  , 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 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 flowinduced 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 ( ∼ 6 • ).In contrast, we focus on large-amplitude (45 • <  < 120 • ) flow-induced oscillations.

Power coefficient map and system stability
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 where 0 ⩽  ⩽ 2.5 rad and 0.15 Hz ⩽   ⩽ 0.6 Hz.The fluid moment   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 5a,  > 700 s).By integrating the governing equation of the passive system 2.2 over  = 20 cycles and taking the cycle average (Zhu et al. 2020), we can get the power coefficient of the system where  0 is the starting time,  is the pitching period and  * =  ∞ / is the non-dimensional time.In this equation, the    * term represents the power injected into the system from the free-stream flow, whereas the  *  * 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   > 0, where the power injected by the ambient flow is higher than that dissipated by the structural damping.On the contrary,   < 0 in the blue regions.The colored dashed lines indicate the   = 0 contours, where the power injection balances the power dissipation, and the system is in equilibrium.
The   = 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,   , are used to calculate the non-dimensional velocity,  * , 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,   , to evaluate  * 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   = 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 4a).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   gradient near   = 0.The junction between the top and the middle   = 0 branches, which corresponds to the saddle-node point, stays relatively sharp for Λ = 0 • -15 • and becomes smoother for Λ = 20 • -25 • .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   = 0 curves for Λ = 0 • -25 • 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 nonmonotonically with Λ, 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   regions as a function of the sweep angle), with an optimal sweep angle at Λ = 10 • , which might inspire future designs of higher efficiency oscillating-foil energy harvesting devices.

Force, moment and three-dimensional flow structures
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,  = 1.05 (60 • ) and  *  = 0.085 (i.e. the black star on figure 6f ), and three sweep angles, Λ = 0 • , 10 • and 20 • .This particular pitching kinematic is selected because it sits right on the   = 0 curve for Λ = 0 • but in the positive   region for Λ = 10 • and in the negative   region for Λ = 20 • (see figure 6a,b,d,f ).
Phase-averaged coefficients of the aerodynamic moment,   , the normal force,   , the tangential force,   , the lift force,   , and the drag force,   , 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 7a) behaves differently for different sweep angles, whereas the shape of other force coefficients (figure 7b,c) does not change with sweep angle, resembling the trend observed in the static measurements (figure 2).The observation that the wing sweep (Λ = 0 • to 25 • ) 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 leadingedge 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)  The collapse of the normal force,   , at different sweep angles suggests that the wing sweep regulates the aerodynamic moment,   , by changing the moment arm,   , as   =     .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,   , peaks at a later time in the cycle, and has an increased maximum value.To further analyze   and its effects on the power coefficient,   , for different wings sweeps, we compare   and   for Λ = 0 • , 10 • and 20 • in figure 7(d-f ), respectively.Note that here we define the power coefficient as   =    * , which is different from equation 3.3 in a way that this   is timedependent instead of cycle-averaged, and that the power dissipated by the structure,  *  * 2 is not considered (this power dissipation is small because a small  * is used in the experiments).The normalized pitching angle, /, and pitching velocity, /(2   ), are also plotted for reference.We see that at the beginning of the cycle (0 ⩽ / < 0.15),   (/) grows near-linearly for all three wings.Because  > 0 for the first quarter cycle, the -intercept of   determines the starting point of the positive   (/) region, corresponding to the left edge of the green panels in the figures.The   > 0 region starts at / = 0 for the unswept wing as   has a near-zero -intercept.For the Λ = 10 • swept wing, because   has a small positive -intercept, the   > 0 region starts even before / = 0. On the contrary, the   > 0 region starts after / = 0 for the Λ = 20 • swept wing due to a small negative -intercept of   .Owing to the combined effect of an increasing   and a decreasing , the power coefficient peaks around / = 0.125 for all the wings.The maximum   of the Λ = 10 • wing is slightly higher than that of the other two wings, due to a slightly higher   .
As the pitching cycle continues,   (/) peaks around / = 0.15, 0.17 and 0.28 for Λ = 0 • , 10 • and 20 • , respectively.The pitch reversal occurs at / = 0.25, where  reaches its maximum and  switches its sign to negative.Because the pitching velocity is now negative, the green panels terminate as   drops below zero, suggesting that   starts to dissipate energy into the ambient fluid.However, because   continues to grow after / = 0.25 for the Λ = 20 • wing, it generates a much more negative   as compared to the wings with a lower sweep angle.Figure 7(a) shows that   decreases faster for the Λ = 10 • wing than the unswept wing at 0.25 ⩽ / < 0.5.This difference results in a less negative   for the Λ = 10 • wing as compared to the Λ = 0 • wing.The faster decrease of   for the Λ = 10 • wing also makes it the first to switch back to positive power generation, where   and  are both negative.The same story repeats after / = 0.5 due to the symmetry of the pitching cycle.In summary, we see that subtle differences in the alignment of   and  can result in considerable changes of   for different sweep angles.The start of the   > 0 region is determined by the phase of   , whereas the termination of the   > 0 region depends on .A non-monotonic duration of the   > 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.
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  1 / = 0.14,  2 / = 0.22 and  3 / = 0.30, and (e-g) the corresponding leading-edge vortex and tip vortex geometries for the Λ = 0 • , 10 • and 20 • wings.The three equally spaced time instants  1 / = 0.14,  2 / = 0.22 and  3 / = 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- surfaces with a value of 50 s −2 and colored by the non-dimensional spanwise vorticity,   / ∞ .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 −axis.
The initial linear growth of the moment coefficient before  1 / for all three wings corresponds to the formation of a strong leading-edge vortex, as depicted in figure 8(bd) at  1 / = 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 8e-g) provides a more quantitative measure to analyze the LEV and TV dynamics.We see that at  1 / = 0.14, the LEVs for all three wings are mostly aligned with the leading edge except for the tip region (/ = 0).For the two swept wings, the LEV also stays closer to the leading edge near the wing root (/ = 3).Due to the high wing sweep of the Λ = 20 • 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   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 Λ = 10 • swept wing, the majority of the tip vortex stays behind the pivot axis.For the Λ = 20 • 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  2 / = 0.22, figure 8(b) and the front view of figure 8(e) show that the LEV mostly 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 Λ = 10 • wing, this small portion of the attached LEV shrinks (see the front view of figure 8f ).The top portion of the LEV near the wing root is also observed to stay attached to the wing surface as compared to the Λ = 0 • case.For the Λ = 20 • 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 Λ = 10 • .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 Λ = 20 • wing, potentially explaining the   of Λ = 20 • exceeding the other two wings at  2 /.The tip vortex moves upwards and outwards with respect to the wing surface from  1 / to  2 /.
During the pitch reversal ( 3 / = 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 Λ = 10 • 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 8f ) as compared to the previous time step.For the Λ = 20 • 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   , 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 6c).
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 Λ = 10 • (figure 8c) and 20 • (figure 8d).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 10c and 13c).Moreover, Visbal & Garmann (2019) reported that for swept wings, as Λ increases, the vortex arch moves towards the wing tip, which is also seen in our experiments (compare the front views of figure 8e-g).

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.
As we discussed in §2.4, the first step of applying FMPM is to construct an 'influence potential', .We solve equation 2.6 numerically using the MATLAB Partial Differential Equation Toolbox (Finite Element Method, code publicly available on MATLAB File Exchange).We use a 3D domain of 10 × 10 × 20, and a mesh resolution of 0.02 on the surface of the wing and 0.1 on the outer domain.We visualize the calculated threedimensional influence field, , for the Λ = 0 • , 10 • and 20 • wings using iso- 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- 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  peaks on the wing surface and decreases towards the far field.The slight asymmetry of  with respect to the pitching axis (see figure 9d) 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- 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 Λ = 10 • and 20 • , we see that the quadrant distribution of the influence field persists.However, the iso- 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  regions extend over the entire chord near the wing root, even behind the pitching axis.Similarly, the positive  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  field for swept wings will result in some non-intuitive distributions of the aerodynamic moment.In addition, the magnitude of the  field is observed to increase with the sweep angle, due to the increase of the effective moment arm (Zhu et al. 2021).
We multiply the three-dimensional  field by the influence field, , and get the spanwise moment (density) distribution field, −2.To visualize the moment distributions, we recolor the same iso- surface plots shown in figure 8 with the moment density, −2, which are shown in figure 10(a-c).As before, the wings and flow fields are rotated by  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- 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  1 / = 0.14 (figure 10a), as expected, we see that the entire LEV on the unswept wing is generating a positive moment.For the Λ = 10 • swept wing, however, the LEV generates a near-zero moment near the wing tip, and for the Λ = 20 • swept wing, the tip region of the LEV contributes a negative moment due to the non-conventional distribution of the  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 −2 over the horizontal (, )-plane at each spanwise location, , we are able to obtain the spanwise distribution of the vorticity-induced moment, shown in figure 10(d-f ).For the unswept wing, Λ = 0 • , figure 10(d) shows that the LEV generates a near-uniform positive moment across the span.As the sweep angle increases (Λ = 10 • ), the LEV generates a higher positive moment near the wing root, and the TV starts to generate a negative moment.For the Λ = 20 • 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 8a).
At  2 / = 0.22 (figure 10b), 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  1 / 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 Λ = 20 • wing generates more positive moments near the wing root as compared to the Λ = 10 • wing due to the magnitude of the  field (figure 9).The TVs for the three wings behave similarly to the cases at  1 /.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 ⩽ / ⩽ 1) generates positive moments.As compared to  1 /, the LEV generates more positive moments near the wing root for the two swept wings, especially for the Λ = 20 • 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 Λ = 20 • wing produces the highest   , followed by the Λ = 10 • wing and then the unswept wing at  2 /.
At  3 / = 0.30 (figure 10c), 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 Λ = 10 • swept wing, although it is largely aligned with the pivot axis (see also the side view of figure 8f ).This is again a result of the non-conventional funnel-shaped  field near the wing root and the teardrop-like  field near the wing tip (figure 9b and e).This trend persists for the Λ = 20 • 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- surface also becomes larger for the Λ = 20 • wing as compared to the previous time steps, indicating a stronger LEV and thus a higher aerodynamic moment, which explains why the   of Λ = 20 • peaks around  3 / 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 Λ = 20 • wing than the Λ = 10 • 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 −2 over the (, )-plane (i.e.figure 10d-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, , is zero near the trailing edge due to symmetry.This means that the contribution to the moment by the TEV is negligible, because −2 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  2 / = 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.

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 • to 25 • .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 (Λ = 10 • ) 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.
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.
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 ∇ 2  = 0, and  n = [( −   ) × n] • e z

Figure 2 .
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.
Amplitude response and static divergence for unswept and swept wings.⊲: increasing , ⊳: decreasing .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 4 .
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 / = 0 corresponds to  = 0.
(a) shows the measured frequency response,  *  , as a function of the calculated natural (structural) frequency,  *  , and sweep angle.In the figure,  *  =   / ∞ and  *  =   / ∞ , where   is the measured pitching frequency and Temporal evolution of (a) the pitching angle , (b) the fluid moment   , and the stiffness moment  *  * near the Hopf point for the Λ = 15 • swept wing.The vertical gray dashed line indicates the time instant ( = 645 s) at which  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,    / *  , versus the measured Hopf point,  *  .The black dashed line shows a 1:1 scaling.
Figure5(a) and (b) shows the temporal evolution of the pitching angle,  (), the fluid moment,   (), and the stiffness moment,  *  * (), for the Λ = 15 • 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 ( < 645 s).The divergence angle is lower than the static stall angle,   , and so we know that the flow stays mostly attached, and the fluid moment,   , is balanced by the stiffness moment,  *  * (figure5b).When the Cauchy number,  = 1/ * , is increased above the Hopf point (figure5a,  > 645 s),  *  * is no longer able to hold the pitching angle below   .Once the pitching angle exceeds   , stall occurs and the wing experiences a sudden drop in   .The stiffness moment,  *  * , 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  *  * (see also figure4b).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  exceeding the static stall angle after  * is decreased below the Hopf point, causing   to drop below  *  * .At this value of  * , the slope of the static stall point should be equal to the stiffness at the Hopf point,  *  (i.e.   =  *   * , where    is the static stall moment).This argument is verified by figure5(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  *   * .We see that the  *   * lines all roughly pass through the static stall points ( *  ,    ) of the corresponding Λ.Note that  *   * of Λ = 15 • and 20 • overlap with each other.Similar to the trend observed for the Hopf point in figure3, the static stall moment    can also be divided into two groups, with the unswept wing (Λ = 0 • ) being in G2 and all the other wings (Λ = 10 • -25 • ) being in G1 (see also figure2b).The inset compares the predicted Hopf point,    / *  , with the measured Hopf point,  *  , 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 thatNegi et al. (2021) performed global linear stability analysis on an

Figure 8 .
Figure 8.(a) Moment coefficients replotted from figure 7(a) for half pitching cycle.Three representative time instants  1 / = 0.14,  2 / = 0.22 and  3 / = 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 Λ = 0 • unswept wing, and the Λ = 10 • and Λ = 20 • swept wings.The flow structures are visualized with iso- surfaces ( = 50 s −2 ) and colored by the non-dimensional spanwise vorticity,   / ∞ .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.

Figure 9 .
Figure 9. Iso-surface plots of three-dimensional influence potentials for (a) the Λ = 0 • unswept wing, (b) the Λ = 10 • swept wing, and (c) the Λ = 20 • 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.

Figure 10
Figure 10.(a-c) Phase-averaged iso- surfaces ( = 50 s −2 ) for the Λ = 0 • unswept wing and the Λ = 10 • and 20 • swept wings, colored by the vorticity-induced moment density, −2 (m 2 s −2 ), at  1 / = 0.14,  2 / = 0.22 and  3 / = 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 −2 at different spanwise locations.