# Rapidly pitching plates in decelerating motion near the ground

Dibya R. Adhikari1<EMAIL_ADDRESS>Samik Bhattacharya1
1Department of Aerospace and Mechanical Engineering, University of Central
Florida, 4000 Central Florida Blvd, Orlando, FL 32816, USA

###### Abstract

Birds employ rapid pitch-up motions for different purposes: perching birds use
this motion to decelerate and come to a complete stop while hunting birds,
like bald eagles, employ it to catch prey and swiftly fly away. Motivated by
these observations, our study investigates how natural flyers accomplish
diverse flying objectives by rapidly pitching their wings during deceleration.
We conducted experimental and analytical investigations focusing on rapidly
pitching plates during deceleration in close proximity to the ground to
explore the impact of ground proximity on unsteady dynamics. Initially, we
executed simultaneous deceleration and pitch-up motion close to the ground.
Experimental results demonstrate that as the pitching wing approaches the
ground, the instantaneous lift increases while the initial peak drag force
remains relatively unchanged. Our analytical model conforms this trend,
predicting an increase in lift force as the wing approaches the ground,
indicating enhanced added mass and circulatory lift force due to the ground
effect. Next, we examined asynchronous motion cases, where rapid pitching
motions were initiated at different stages of deceleration. The results reveal
that when the wing pitch is synchronized with the start of deceleration,
larger counter-rotating vortices form early in the maneuver. These vortices
generate stronger dipole jets that orient backward in the later stages of the
maneuver after impinging with the ground surface, which hunting birds recover
to accelerate after catching prey. Conversely, when the wing pitch is delayed,
smaller vortices form, but their formation is postponed until late in the
maneuver. This delayed vortex formation generates beneficial unsteady forces
late in the maneuver that facilitates a smooth landing or perching. Thus,
through strategic tuning of rapid pitch-up motion with deceleration, natural
flyers, such as birds, achieve diverse flying objectives.

###### keywords:

Perching wing, hunting, rapid area change, ground effect, dipole jet,

##  1 Introduction

Birds have been observed to pitch their wings for a variety of purposes: such
as perching birds that rapidly pitch their wings upward to decelerate to a
complete stop while landing (Carruthers et al., 2007, 2010; Berg & Biewener,
2010; Provini et al., 2014), and hunting birds, which also use the same motion
to slow down to catch a fish out of water before flying away (Reimann, 1938;
Todd et al., 1982; Venable, 1996; Stalmaster & Kaiser, 1997; Gerrard &
Bortolotti, 2014; Sörensen, 2015; Collard III & Brickman, 2021). A review of
existing research revealed that there are several basic aerodynamic questions
related to pitching plates in deceleration that have not been sufficiently
addressed in the current literature. These include: (1) how ground proximity
affects the unsteady dynamics of the perching maneuver, and (2) how birds
achieve different flying objectives by rapidly pitching their wings during
deceleration. In this paper, we aim to investigate these questions using
experimental and analytical approaches.

Reducing the distance between the wing and the ground increases the lift-to-
drag ratio due to the ground effect (Zerihan & Zhang, 2000; Luo & Chen, 2012).
The use of the ground effect has been observed in many natural flyers and
swimmers (Baudinette & Schmidt-Nielsen, 1974; Hainsworth, 1988; Withers &
Timko, 1977; Blake, 1979; Webb, 1993; Saffman, 1967; Park & Choi, 2010), which
have evolved to take advantage of the ground effect to enhance their
performance. This increase in operational efficiency has inspired the design
of wings in ground effect aircraft (Rozhdestvensky, 2006).

Numerous efforts have been made in recent decades to understand the ground
effect experienced by an oscillating airfoil close to the ground (Fernández-
Prats et al., 2015; Mivehchi et al., 2016; Zhang et al., 2017). Studies have
revealed that the effects of ground proximity on unsteady aerodynamics can
vary depending on the type of wing kinematics. For instance, in the study by
Quinn et al. (2014) on a pitching airfoil, they found that when the airfoil is
close to the ground, it experiences increased lift force that pushes the
airfoil away from the ground. They also observed that pitching near the ground
generates a vortex pair instead of a vortex street, increasing the average
thrust force. In another study, Deepthi & Vengadesan (2021) showed that an
inclined flapping wing in ground effect experiences a higher lift at a stroke
plane angle of $45^{\circ}$, but not at other angles. Therefore, to fully
quantify the performance of perching birds, further exploration of the ground
effect experienced by a rapid pitching plate during deceleration is necessary.

Rapid pitching causes a quick change in the surface area of the wing facing
the airflow, which can significantly impact the airflow over the wing. This
rapid change in the wing’s surface area can have fascinating prospects for
flow control, as the added mass changes with the change in the surface area,
which can play a significant role in varying the dynamic forces on the wing.
Saffman (1967) showed that a body could propel itself by deforming its surface
area through added mass recovery. Childress et al. (2006) conducted an
experimental study on a flexible body in oscillating air. They concluded that
exposing the variable frontal area to the airflow due to wing flapping leads
to changes in the added mass, resulting in stable hovering. Rapid area change
can lead to boundary layer separation and the shedding of the vortices on a
vanishing body (Wibawa et al., 2012). Weymouth & Triantafyllou (2013)
concluded that in addition to added mass recovery, the deforming body relies
on flow separation elimination to achieve ultra-fast escape. However, our
understanding of rapid area change is limited to decreasing surface area in
acceleration, and the study of how the flow behaves over an increased surface
area during deceleration, which is commonly experienced by birds that perch
and hunt, has yet to be extensively studied.

Recently, Polet et al. (2015) conducted an experimental and numerical study on
the unsteady aerodynamics of a two-dimensional NACA0012 airfoil undergoing
simultaneous pitch-up and decelerating motion. They found that the significant
lift and drag force on a wing during perching is mainly caused by the added
mass effect and the formation of strong vortices at the leading and trailing
edge of the wing. Jardin & Doué (2019) also performed a numerical study on a
perching airfoil and concluded that a minimum kinetic energy could be achieved
on the airfoil at the end of the perching maneuver at a higher pitch rate or
the lift and drag force on the airfoil can be enhanced by increasing the pitch
rate. Adhikari et al. (2022) also studied the unsteady dynamics of a finite
wing undergoing a rapid pitch-up motion while decelerating and descending
close to the ground. They showed that a perching wing could generate higher
forces by using a combination of pitching and heaving motions during
deceleration. However, in these studies, although the wing generated a higher
drag force by increasing the pitch rate, which is appropriate for decelerating
rapidly to a complete stop, the perching wing also generated a higher lift
force. This higher lift force causes the wing to rise in altitude, which may
not be desirable for perching at the initial perching location or altitude.
Thus, more research is needed to understand the mechanics involved when the
wing pitches rapidly during deceleration.

Studies on rapid area change have also investigated the effect of varying the
frontal area against the incoming airflow. Spagnolie & Shelley (2009) found
through a numerical simulation that by controlling the phase difference
between the shape change and background flow of an oscillating flow, a shape-
changing body can generate vortex structures that induce a downward moving
dipole jet below the body. The resulting jet of fluid enabled the body to
hover or ascend vertically. Similarly, Weymouth & Triantafyllou (2013) showed
that, by deforming quickly, a rapidly shape-changing body could eliminate the
flow separation from its surface. This reduces drag and increases thrust
force, which is beneficial for escape maneuvers. While these studies provide
insight into the timing of the shape change with the airflow and its impact on
the added mass and vortex evolution, they were limited to oscillating or
accelerating flows. Therefore, this paper focuses on varying the timing of
rapid pitch-up motion relative to decelerating flow and assesses the impact of
such motion on the perching and hunting birds’ performance.

In this paper, we consider two scenarios of rapid pitch-up motion of plates in
decelerating motion near the ground to better understand the aerodynamic
mechanism used by perching and hunting birds. In the first scenario, we
perform a synchronous motion where the wing pitches up while decelerating from
steady velocity to a complete stop, with the same motion duration for both
motions. We execute synchronous motion at different ground heights to
understand the ground effect on perching plates. In the second scenario, we
create an asynchronous motion where the deceleration time is longer than the
pitch time of the plate, allowing the execution of the pitch-up motion at
various stages of deceleration. By comparing the evolution of unsteady forces
and flow field while varying the start of pitch-up motion during deceleration,
the current study aims to gain new insights into the aerodynamic mechanisms
natural flyers use and how these mechanisms can be replicated in the design of
next-generation flying vehicles and aircraft.

Figure 1: Schematic diagram of the experimental set-up illustrating the
rectangular plate, with details on the placement of force sensor, camera, and
laser.

##  2 Methodology

###  2.1 Experimental setup

The schematic diagram of the experimental setup is shown in figure 1. The
tests were conducted in a water-filled towing tank with a free surface
measuring $0.9\text{\,}\mathrm{m}$ in length, $0.45\text{\,}\mathrm{m}$ in
width, and $0.4\text{\,}\mathrm{m}$ in height. The wing model was mounted on a
liner stage powered by a servo motor (FSL120, FUYU Inc., China), which moved
it along the length of the towing tank. The wing model’s deceleration was
prescribed by gradually slowing down the linear stage. A stepper-driven linear
stage (LSQ150B-T3, Zaber Tech. Inc., Canada) connected orthogonally to the
servo-driven stage moved the wing model towards the solid boundary, which
acted as a ground in this study. A rotary stage powered by a stepper motor
(RSW60A-T3, Zaber Tech. Inc., Canada) executed the rapid pitch-up motion
around the mid-chord of the wing model. A force sensor was installed on the
setup below the pitching motor and was connected to the wing model via a
$0.09\text{\,}\mathrm{m}$ thick cylindrical rod. A pulse generator (9400
series, Quantum Composers Inc., USA) sent a trigger pulse signal to
synchronize the deceleration motion, the start of the pitching motor, the
force sensor, and the camera. The wing model was submerged vertically in the
tank, with the wingtip positioned $0.2\text{\,}\mathrm{m}$ from the bottom.

Nomenclature |  0$\dot{U}\hphantom{0}$\text{\,}\mathrm{(}\mathrm{m}\mathrm{/}\mathrm{s}^{2}\mathrm{)}$$ | $\dot{\alpha}\hphantom{0}$\text{\,}\mathrm{(}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{/}\mathrm{s}\mathrm{)}$$ | $\Xi$ | Synchronous | Asynchronous ($t^{*}_{os}$)
---|---|---|---|---|---
00000C1 | 0.0351 |  00.5500 | 0.2 | ✓ | $\times$
0000C2_0 | 0.0234 |  00.5500 | 0.3 | $\times$ | ✓$(0.00)$
0000C2_45 | 0.0234 |  00.5500 | 0.3 | $\times$ | ✓$(0.25)$
0000C2_90 | 0.0234 |  00.5500 | 0.3 | $\times$ | ✓$(0.50)$
00000C3 | 0.0877 |  01.3770 | 0.4 | ✓ | $\times$
0000C4_0 | 0.0585 |  01.3770 | 0.6 | $\times$ | ✓$(0.00)$
0000C4_45 | 0.0585 |  01.3770 | 0.6 | $\times$ | ✓$(0.25)$
0000C4_90 | 0.0585 |  01.3770 | 0.6 | $\times$ | ✓$(0.50)$
00000C5 | 0.1111 |  01.7450 | 0.6 | ✓ | $\times$
0000C6_0 | 0.0741 |  01.7450 | 0.9 | $\times$ | ✓0$(0.00)$
0000C6_45 | 0.0741 |  01.7450 | 0.9 | $\times$ | ✓$(0.25)$
0000C6_90 | 0.0741 |  01.7450 | 0.9 | $\times$ | ✓$(0.50)$
Table 1: Summary of the kinematic parameters

Figure 2: (a) Comparison of the variation of non-dimensional velocity $U^{*}$
between synchronous and asynchronous motions. Variation of velocity and angle
of attack for two perching scenarios: (b) synchronous motion; and (c)
asynchronous motion. Comparisons shown are represented as a function of non-
dimensional time $t^{*}=t/t_{p}$, where $t_{p}$ is the time period of pitch-up
motion. Here, the decelerating velocity is scaled by the steady state
velocity, $U_{\infty}$, while the total change in the angle, which is
$90^{0}$, scales the angle of attack during pitch-up motion. The ratio of the
time period of deceleration between synchronous, $t_{{d}_{s}}$, and
asynchronous motion, $t_{{d}_{as}}$, is
$t_{r}=\frac{t_{{d}_{s}}}{t_{{d}_{as}}}=\frac{1}{1.5}$. For the synchronous
case, $t_{{d}_{s}}=t_{p_{s}}$, whereas for asynchronous case,
$t_{{d}_{as}}=1.5*t_{p_{as}}$.

###  2.2 Wing model and problem description

We used a finite rectangular wing planform with a chord length ($c$) of
$0.05\text{\,}\mathrm{m}$ and a planform area of
$0.0075\text{\,}\mathrm{m}^{2}$ as a perching wing model. The aspect ratio
($AR$) of the wing was $3\text{\,}$ and was fabricated from
$6\text{\,}\mathrm{m}\mathrm{m}$ thick flat aluminum plate. The wing’s leading
edge (LE) was rounded, and the trailing edge (TE) was sharpened to meet the
Kutta condition, ensuring the flow smoothly leaves the TE.

To simulate the perching maneuvers, two scenarios were considered: synchronous
motion and asynchronous motion. In both scenarios, the wing model was
initially oriented at an angle of attack $\alpha_{0}=0^{0}$ and then rapidly
pitched up to $\alpha=90^{0}$ while undergoing deceleration.

In synchronous motion, the wing decelerated from steady velocity $U_{\infty}$
of $0.1\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ to a complete stop while
pitching up, with both motions having the same motion duration
($t^{*}_{decn}=t^{*}_{pitch}$; see Figure 2 (b)). This means that the start
and end of the deceleration and rapid pitch-up motions are synchronized. The
pitch-up motion causes a rapid increase in the frontal area of the wing facing
the flow. This increase, combined with simultaneous deceleration, was
quantified using the shape change number:

$\Xi=\frac{V}{\triangle U}$ (1)

where $V$ is the frontal area speed of the airfoil and $\triangle U$ is the
change in the translation speed of the wing during deceleration. We executed
three shape change numbers ($\Xi$ = 0.2, 0.4, $\&$ 0.6) at ten non-dimensional
ground heights ranging from $h^{*}$ = $\frac{h}{c}$ = $1.5-0.04$. We refer to
$h^{*}$ = 1.5 as far from the ground case and $h^{*}$ = 0.04 as close to the
ground case.

In asynchronous motion, the deceleration time was extended compared to
synchronous (see Figure 2 (a)), while keeping the pitching time constant. This
results in time offsets between the two motions, with the deceleration time
longer than the time to pitch, i.e. $t^{*}_{decn_{as}}$ =
$1.5*t^{*}_{pitch_{as}}$. This time offset allows the pitch-up motion to be
executed at various stages of the deceleration. As a result, for the same
pitch rate, the change in velocity when the wing completes the pitch-up motion
is $\triangle
U=U_{\infty}*\frac{t^{*}_{pitch}}{t^{*}_{decn}}=U_{\infty}*\frac{1}{1.5}$,
resulting in a higher shape change number $\Xi$ for asynchronous motion
compared to synchronous case, i.e. $\Xi_{asyn}=1.5*\Xi_{syn}$. For each $\Xi$,
we considered three starting time offsets ($t^{*}_{os}$ = 0, 0.25, and 0.5)
between the decelerating and pitch-up motion. When $t^{*}_{os}=0$, the start
of the deceleration and pitch-up motions are in sync, but the pitch-up motion
ends before the wing decelerates to a complete stop. With $t^{*}_{os}=0.5$,
the start of the pitch-up motion lags the start of the deceleration motion,
but the end of the pitch-up motion and deceleration motion is synchronized.
Each asynchronous motion case was executed at three non-dimensional ground
distance $h^{*}$ = 1.5, 0.25, and 0.04.

The steady-state velocity of the wing model was ($U_{\infty}$) of
$0.1\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$. The Reynolds number of the
perching wing model, based on $U_{\infty}$ and $c=$0.05\text{\,}\mathrm{m}$$,
was $Re=6,500$. Table 1 summarizes the kinematic parameters used in this
experiment.

###  2.3 Measurement of Instantaneous forces

We measured the instantaneous forces acting on the wing using a six-axis force
and torque sensor (NANO 17, ATI Inc., USA) connected to a 16-bit DAQ device
(NI-USB-6211, National Instrument, USA). The force-sensor data was collected
at a sampling rate of $5\text{\,}\mathrm{kHz}$ and averaged over five test
runs. The combined wing motion produced an oscillatory frequency of
approximately $4\text{\,}\mathrm{H}\mathrm{z}$ on the force-sensor data. We
filtered the force-sensor data with a Butterworth low pass filter with a
cutoff frequency of $3\text{\,}\mathrm{H}\mathrm{z}$ to remove this vibration,
while retaining most of the fluid force oscillatory peaks. Subsequently, we
smoothed the data using the moving average of 20 points. We found that the
uncertainty in the force data is found to be around $6\%$ at the peak and less
than $3\%$ for the smaller magnitude of the forces.

To account for the inertial forces and the weight of the wing assembly, we
performed both dynamic and static tare experiments. In dynamic tare, we
conducted tare experiments in the air using the same kinematics as in the
water. We observed that the lift force in water was approximately 11 times
higher than in the air. Since the apparent mass of the water accelerated along
the model was approximately 10 times higher than the mass of the model and the
force balance, we neglected the dynamic tare in the air, following the
approach proposed by Barlow et al. (1999) and Granlund et al. (2013). In
static tare, we measured the data in still water every $3^{\circ}$ of the
pitch angle up to the maximum pitch angle. The wing model produced negligible
static tare, so its contribution was not considered in our analysis.

###  2.4 PIV measurements

We measured the velocity field at the 50% wing span using Planar particle
image velocimetry (PIV). To seed the water tank, we used neutrally buoyant,
$10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter silver-coated hollow glass
spheres (Conductophill, Potter Inc., USA). The laser sheet for illuminating
the plane of interrogation was generated by a $2\text{\,}\mathrm{m}\mathrm{m}$
diameter beam from a continuous-wave green laser (DPSS-DMPV-532-2, Egorov
Scientific, USA), which was expanded into a $2\text{\,}\mathrm{m}\mathrm{m}$
thick laser sheet by using two cylindrical lenses. We recorded images of the
illuminated plane by a AOS camera (J-Pri, AOS Tech. AG, Switzerland) at a
frame rate of $200\text{\,}\mathrm{Hz}$ and a resolution of 2560 x 1920
pixels. The field of view was $0.25\text{\,}\mathrm{m}$ x
$0.18\text{\,}\mathrm{m}$. The images were processed in PIVLab, a MATLAB-based
software. We used a multi-pass iterative algorithm with a window size of 64 x
64 pixels in the first pass and 32 x 32 pixels in the second pass, with a
$50\%$ overlap between successive windows. Finally, we phase-averaged the PIV
data over five runs.

##  3 Results and Discussion

In § 3.1 and § 3.2, we present the results separately for synchronous and
asynchronous motion, where synchronous motion refers to cases where the start
and end of deceleration and pitch-up motions are in synchrony, while
asynchronous motion refers to the cases where the two motions are not in
synchrony. We then discuss the dipole jet induced due to counter-rotating
vortices in § 3.3. Next, in § 3.4, we focus on the scaling laws for perching
maneuvers. Finally, in § 3.5, we compare instantaneous forces between the
experimental and analytical model results.

Figure 3: Comparisons of (a) lift and (b) drag coefficient on the rectangular
wing for $\Xi$ = 0.2, 0.4, and 0.6. Each $\Xi$ are presented at four non-
dimensional ground height, $h^{*}$.

###  3.1 Synchronous motion: Unsteady forces and flow-field

Figure 3 displays the evolution of unsteady lift and drag forces during
synchronous motion at three different shape change numbers $\Xi=$ 0.2, 0.4,
and 0.6. To provide a comprehensive analysis of the ground effect, we executed
each $\Xi$ at ten different non-dimensional ground heights ranging from
$h^{*}=1.5-0.04$. For a clear and concise representation of the plot, we
provided unsteady forces at four different values of $h^{*}$. The deceleration
and pitch-up motion were both initiated at $t^{*}=0$, and the unsteady forces
were measured during perching.

Figure 3(a) shows that the execution of simultaneous deceleration and pitch-up
motion results in a steep rise in the lift coefficient and attains the peak
value after a certain time instant. This initial rise in the lift coefficient
is mainly due to the combined effect of non-circulatory and circulatory force.
The plot indicates that the peak lift force coefficient increases with
increasing $\Xi$, which is consistent with the results of Polet et al. (2015)
and Jardin & Doué (2019). The peak lift coefficient increases by approximately
50% as $\Xi$ increases from 0.2 to 0.6. After this initial peak force, the
wing experiences a decline in the lift coefficient. This decay in the lift is
related to the detachment of the leading edge vortex (LEV) from the wing
leading edge (LE), leading to the stalled flow at the latter stage of the
motion. This decline in lift coefficient can also be related to the decrease
in the non-circulatory force due to the deceleration of the wing. From figure
3 (a), it is observed that the lift force for $\Xi=0.6$ starts to decay at a
later stage of the motion compared to that of $\Xi=0.2$.

Figure 3 (a) also illustrates the effect of ground proximity on the
instantaneous lift coefficient of the perching plates at various non-
dimensional ground heights ranging from $1.5\leq h^{*}\leq 0.04$. As $h^{*}$
decreases, the initial rise in the lift force increases consistently for each
$\Xi$. For $\Xi=0.2$, the initial peak lift force rises by approximately
$19\%$, whereas for $\Xi=0.6$, this rise is approximately $38\%$. Although the
initial peak force increases with ground proximity, the perching plate also
experiences an increase in the trough force at the end of the maneuver as the
wing approaches the ground. However, for the majority of the perching
maneuver, the instantaneous life force increases when the wing is close to the
ground, which can be beneficial during this highly unsteady maneuver.

Figure 4: Contours of the normalized vorticity field, $\omega^{*}$, for
synchronous motion at the $50\%$ of the wing span at three time-steps, $t^{*}$
= 0.37, 0.62, & 0.85: $\Xi$ = 0.2 at (a) $h^{*}$ = 1.5; (b) $h^{*}$ = 0.04.
$\Xi$ = 0.6 at (c) $h^{*}$ = 1.5; (d) $h^{*}$ = 0.04.

The evolution of the drag coefficient on the perching plate is shown in figure
3 (b). It is observed that higher values of $\Xi$ lead to a larger
instantaneous drag coefficient, with the peak drag coefficient increasing by
approximately $10\%$ when $\Xi$ is changed from 0.2 to 0.6. Interestingly, the
time instant of the initial peak drag force shifts from $t^{*}=0.45$ for
$\Xi=0.2$ to $t^{*}=0.52$ for $\Xi=0.6$. This trend is similar to that
observed for the lift coefficient, which suggests that the peak and decay of
the forces occur at a higher angle of attack for higher values of $\Xi$. This
phenomenon has also been observed by KleinHeerenbrink et al. (2022), who
concluded that perching birds pitch faster to minimize the stall distance.

In contrast to the lift, the evolution of the drag coefficient is not
significantly affected by ground proximity. For each $\Xi$, varying $h^{*}$
leads to a negligible change in the initial peak drag force. However, the
ground effect does impact the parasitic thrust, which increases for all $\Xi$
values when the wing is close to the ground.

To better understand our findings, we analyze the vorticity field for $\Xi$ =
0.2 & 0.6 at two extreme ground heights, i.e., $h^{*}$ = 1.5 & 0.04. Figure 4
presents the normalized vorticity fields at time instances, $t^{*}$ = 0.37,
0.62, & 0.85, which highlight the key changes in the flow field due to
variations in $\Xi$ and ground height. To normalize the vorticity field, we
use the wing’s chord and the steady-state translational velocity, $\omega^{*}$
$=\frac{\omega*c}{U_{\infty}}$.

Our PIV results demonstrate that a rapid pitch-up motion during deceleration
causes the shear layer to separate, leading to the formation of counter-
rotating leading edge vortex (LEV) and trailing edge vortex (TEV) structures
(Figure 4). Although both values of $\Xi$ result in similar vortex formation,
smaller $\Xi$ produces larger LEV that diffuses faster and is farther away
from the plate surface, whereas higher $\Xi$ leads to more coherent and
stronger vortex structures closer to the wing. When the plate is pitching
slowly while decelerating slowly, this motion generates a smaller pressure
gradient on the wing surface, which creates weaker vortices that are more
spread out. However, rapid pitch-up motion during rapid deceleration induces a
large pressure gradient due to the rapid change in the flow direction and
velocity, leading to the formation of stronger and more coherent vortex
structures closer to the wing. The stronger and more coherent vortex closer to
the wing surface induces more impulse on the wing than the vortices that are
weaker and more spread out, explaining the larger value of lift and drag force
observed for $\Xi$ = 0.6 compared to $\Xi$ = 0.2.

For both $\Xi$, at $t^{*}$ = 0.37 and 0.62, the TEV is larger and stronger at
$h^{*}$ = 0.04 than at $h^{*}$ = 1.5, which is especially evident for $\Xi$ =
0.6. The proximity of the plate to the ground constrains the flow around the
plate, leading to an increase of pressure below the wing (Ahmed & Sharma,
2005), which causes the flow to curl more strongly around the edges of the
plate (Lee & Ko, 2018), resulting in the larger and stronger TEV as seen in
the near ground case. A stronger TEV, in turn, induces stronger velocities on
the LEV, bringing them closer together. Wu et al. (1998) found that this close
vortex pair induces a stronger downwash. For $\Xi$ = 0.6 at $t^{*}$ = 0.62, a
stronger dipole jet oriented downward is generated in the near ground case,
producing an upward force, which can explain the higher value of lift force
observed at $h^{*}$ = 0.04 compared to $h^{*}$ = 1.5.

The shed vortices near the ground experience additional shear stress from the
ground due to the no-slip condition at the boundary. This shear stress
prevents the shed vortices from spreading out, resulting in more compact shed
vortices (see figure 4 (d)). The compact shed vortices can reduce the adverse
pressure gradient on the wing surface and delay the formation of secondary
vortices, enhancing the stability of the LEV and increasing lift (McLean,
2012).

However, at the end phase of the maneuver, when the wing is close to the
ground, the dipole jet gets impinged to the ground. At $t^{*}$ = 0.85, figure
4 shows that this impingement advects the shed LEV and TEV further apart. For
$\Xi$ = 0.6, the x-distance between the LEV and TEV is 0.92c for the near
ground case compared to 0.76c for the far ground case, resulting in a lower
impulse on the wing. This results in lower impulse on the wing. This may
explain the increased drop in the lift and drag force at the end of the
maneuver on the perching wing close to the ground.

Figure 5: Comparisons of the instantaneous (a) lift and (b) drag coefficient
on the rectangular wing for $\Xi$ = 0.3, 0.6, and 0.9. Each $\Xi$ is presented
at three-time offsets between the decelerating and pitch-up motion.

###  3.2 Asynchronous motion: Unsteady forces and flow-field

To investigate the influence of varying frontal area expansion on decelerating
flight during perching, we employed asynchronous motion with three starting
time offsets, $t^{*}_{os}$ = 0, 0.25, and 0.5 between the decelerating and
pitch-up motions. Figure 5 (a) and (b) presents the evolution of lift and drag
coefficient on the rectangular plate for three values of $\Xi$ and three
values of $t^{*}_{os}$. Note that in the asynchronous case, the pitch rate is
the same as in the synchronous case, but we reduced the deceleration value,
resulting in an increase in $\Xi$ by a factor of $\frac{1}{a}$.

From figure 5 (a), it is observed that increasing $\Xi$ enhances the
instantaneous lift coefficient, similar to the behavior observed in the
synchronous case. For $t^{*}_{os}$ = 0, although the pitch rate is the same
for both synchronous and asynchronous motions, the asynchronous motion
produces a peak lift coefficient approximately $40\%$ higher due to the higher
translational velocity experienced by the pitching plate. This higher velocity
enhances both the circulatory and non-circulatory forces. However, for higher
time offsets $t^{*}_{os}$ = 0.25 and 0.5, the lift coefficient starts to rise
later and generates a lower peak lift coefficient compared to $t^{*}_{os}$ =
0. This delay in the rise of the lift coefficient is consistent with the
delayed start of the pitch-up motion and the reduction in the lift coefficient
can be correlated with the lower translational speed caused by the
deceleration of the wing. These trends are consistent for all $\Xi$ considered
in this study, with smaller $\Xi$ resulting in a lower peak lift coefficient.

Varying the starting time offsets between the deceleration and pitch-up motion
leads to a distinct evolution of instantaneous lift coefficients. For
$t^{*}_{os}$ = 0, the perching plate generates a high initial lift force but
also experiences a rapid drop-off in the lift, which may reduce the control
authority of landing birds. However, delaying the rapid pitch-up motion until
late in the deceleration can delay the drop-off of lift force, allowing the
wing to generate lift at the end of the motion and enhance control authority
during this highly unsteady maneuver.

The evolution of the drag coefficient for asynchronous motion is presented in
figure 5 (b). At $t^{*}_{os}$ = 0, each value of $\Xi$ exhibits an increase in
the peak drag coefficient compared to the synchronous case. As the starting
time offset between the two motions increases, there is a reduction in the
peak drag coefficient. Although the peak drag force is reduced at $t^{*}_{os}$
= 0.5, the perching plate generates a positive drag force for the majority of
the flight, as opposed to negative drag or parasitic thrust in the latter
stage of the maneuver for $t^{*}_{os}$ = 0. Furthermore, as we increase
$t^{*}_{os}$, we observe a delay in the onset of parasitic thrust generation
and a decrease in its magnitude. These results suggest that perching birds may
have better control over the aerodynamic forces during landing at higher
$t^{*}_{os}$ due to continuous drag generation and reduced parasitic thrust.

Figure 6: Comparisons of the lift and drag coefficient on the rectangular wing
for $\Xi$ = 0.9 at three non-dimensional ground height, $h^{*}$.

We further analyzed the influence of asynchronous motion by plotting the time-
averaged lift and drag coefficients in figure 5 (c & d). Increasing the
starting time-offset, $t^{*}_{os}$, results in a decrease in the time-averaged
lift coefficient, $C_{L_{avg}}$ as shown in figure 5 (c). For $\Xi$ = 0.9, the
$C_{L_{avg}}$ decreases from 0.7 to near zero as $t^{*}_{os}$ increases from 0
to 0.5. While the drag plot in figure 5 (d) also shows a reduction in the
time-averaged drag coefficient, $C_{D_{avg}}$, with increasing $t^{*}_{os}$,
the decrease in drag is small compared to that of the $C_{L_{avg}}$. For $\Xi$
= 0.9, the $C_{D_{avg}}$ decreases from 0.52 to 0.4 as $t^{*}_{os}$ increases
from 0 to 0.5. The resulting near-zero lift force and positive drag force can
help birds perch on the original landing or perching location without gaining
altitude, providing a beneficial perching strategy.

Figure 6 shows the behavior of unsteady forces for $\Xi$ = 0.9 at three non-
dimensional ground heights: $h^{*}$ = 1.5, 0.5, and 0.04. As the perching
plate approaches the ground, the instantaneous lift coefficient increases
similarly to synchronous cases, and this behavior is consistent across all
starting time offsets. The result indicates that lower ground height provides
more lift enhancement, with the greatest benefits observed at $t^{*}_{os}$ =
0. In contrast, ground proximity has less impact on drag force at the early
stage of the maneuver (see figure 6), but once the perching plate attains its
peak at $t^{*}_{os}$ = 0, the drag force drops rapidly for near-ground case,
creating higher negative drag force or parasitic thrust force at the end of
the maneuver. While a similar drop in drag force is observed at $t^{*}_{os}$ =
0.5, this decrease in drag force is relatively small compared to the drop at
$t^{*}_{os}$ = 0. This indicated that introducing a time offset could be the
optimum way to execute a perching maneuver during landing, as it helps reduce
the risk of losing control authority over aerodynamic forces.

Figure 7: Contours of the normalized vorticity field, $\omega^{*}$, for
asynchronous motion at the $50\%$ of the wing span for $\Xi$ = 0.9 at three
time-steps, $t^{*}$ = 0.5, 1.0, & 1.5: $t^{*}_{os}$ = 0 at (a) $h^{*}$ = 1.5;
(b) $h^{*}$ = 0.04. $t^{*}_{os}$ = 0.5 at (c) $h^{*}$ = 1.5; (d) $h^{*}$ =
0.04.

We investigated the normalized vorticity field to further analyze the results
observed in the asynchronous motion. Specifically, we focused on the highest
shape change number, $\Xi$ = 0.9, and examined two starting time-offset cases,
$t^{*}_{os}$ = 0 & 0.5, corresponding to executing rapid pitching at different
deceleration stages. This allowed us to highlight the key change in the
evolution of flow physics in asynchronous motion. Figure 7 displays the
normalized vorticity field at three-time instants, $t^{*}$ = 0.5, 1.0, and
1.5, and at two extreme ground height cases, $h^{*}$ = 1.5 and 0.04.

Examining figure 7 for $t^{*}_{os}$ = 0, we observed that at $t^{*}$ = 0.5,
the pitching plate generates a coherent leading-edge vortex (LEV) and
trailing-edge vortex (TEV). These stronger and more coherent vortex structures
produce higher lift and drag forces, which is consistent with the evolution of
higher unsteady forces observed in the instantaneous force plot for
$t^{*}_{os}$ = 0. However, at the same time instant, the vorticity field for
$t^{*}_{os}$ = 0.5 reveals the development of the wake downstream of the
plate. This wake opposes the motion of the wing and contributes to the
generation of drag force, explaining the early stage generation of drag force
in $t^{*}_{os}$ = 0.5 cases. Furthermore, as seen in the synchronous case, the
vortices are stronger at $h^{*}$ = 0.04 than at $h^{*}$ = 1.5, explaining the
higher lift force observed at $h^{*}$ = 0.04 compared to $h^{*}$ = 1.5.

Meanwhile, at $t^{*}$ = 1.0, the vortices are fully developed and shed from
the wing surface for $t^{*}_{os}$ = 0 cases, but for $t^{*}_{os}$ = 0.5 cases,
the vortices are still growing. This difference in the vortex development
explains why the unsteady forces drop for $t^{*}_{os}$ = 0 cases but continue
to increase for $t^{*}_{os}$ = 0.5 cases at $t^{*}$ = 1.0. In the former case,
stronger and more coherent vortex structures developed earlier in the maneuver
induce a stronger dipole jet, which impinges on the grounds and separates the
shed vortices further apart, as seen in figure 7 (b) at $t^{*}$ = 1.0 and
$t^{*}$ = 1.5. At $t^{*}$ = 1.5, for $t^{*}_{os}$ = 0 case close to the
ground, the x-distance between the LEV and TEV is 1.24c, contributing to the
more pronounced drop-off of the unsteady force. In contrast, for $t^{*}_{os}$
= 0.5 cases, the weaker vortices result in a slower jet and closer separation
between the shed vortices (0.6c) and the wing surface, resulting in a less
pronounced drop-off of the unsteady force.

###  3.3 Discussion on Dipole Jet

Figure 8: Schematics showing the evolution of the dipole jet and its
interaction with the ground.

This research aims to investigate the mechanics of rapidly pitching plates
during deceleration near the ground to understand how the perching and hunting
birds achieve their impressive maneuverability. By executing rapid pitching at
different stages of deceleration, such as when the forward translational
velocity is still high versus when it is low, we observe distinct changes in
the evolution of vortex structures and forces acting on the plate.

Executing a rapid pitch-up motion during deceleration causes the shear layer
to separate, forming a counter-rotating leading edge vortex (LEV) and trailing
edge vortex (TEV) vortex structures, as shown by our PIV results. The pitch-up
motion combines the counter-rotating vortices to form a comoving vortex
dipole. Once a vortex dipole is created, they will interact with each other,
creating a region of high vorticity gradient. This region acts as a source of
fluid, moving fluid outward, supplying fluid to the vortices needed to
maintain their coherent structures, and forming the jet flow of the dipole. At
the same time, the region between the two vortices experiences a low-pressure
zone due to the centrifugal forces generated by the counter-rotating vortices.
This low-pressure region acts as a sink flow, drawing fluid between the two
vortices. This combination of source flow and sink flow forms a dipole jet
that is characteristic of counter-rotating vortices. The schematic diagram in
figure. 8 illustrates the pitch-up motion of the wing and the resulting
formation of the dipole jet. This dipole jet moves a considerable amount of
momentum carrying fluid with it, which can be used to generate unsteady forces
on the wing.

Figure 9: Velocity field for close to ground case, $h^{*}$ = 0.04. (a, b, c)
$t^{*}_{os}$ = 0; (d, e, f) $t^{*}_{os}$ = 0.5. (a, d) $t^{*}$ = 0.5; (b, e)
$t^{*}$ = 1; (c, f) $t^{*}$ = 1.5.

Figure 9 displays the dipole jet formation during the asynchronous motion by
analyzing the velocity field at two different starting time-offsets cases,
$t^{*}_{os}$ = 0 & 0.5. For $t^{*}_{os}$ = 0 case, pitch-up motion is executed
at a high forward translational velocity, resulting in larger, stronger
counter-rotating vortices and a more significant induced dipole jet (as seen
in figure 9 (a)). This dipole jet is initially directed downward and forward,
producing lift and drag forces. However, when the wing pitches up to its final
effective angle of attack, i.e., at $t^{*}$ = 1, the dipole jet gets deflected
on the ground surface. This deflected jet moves the vortex pair apart,
reducing the impulse they generate on the wing and causing a rapid drop in the
lift and drag force at the end phase of the maneuver. At $t^{*}$ = 1.5, some
deflected jets are reversed and oriented backward, producing a parasitic
thrust force. Weymouth & Triantafyllou (2013) showed that to achieve the
ultra-fast escape, the deforming body stores added mass energy in the fluid
during the early phase of the maneuver by deforming. This energy is then
recovered in the later phase of the maneuver to accelerate the deformed body.
In the present study, the pitching plates recover the deflected jet, which is
oriented backward in the later phase of the motion, as parasitic thrust.
Hunting birds like eagles can use this parasitic thrust to accelerate after
catching their prey.

On the other hand, when executing a pitch-up motion at a low forward
translational velocity ($t^{*}_{os}$ = 0.5), smaller and weaker vortex
structures are formed, which induces a slower dipole jet that moves a lower
amount of momentum-carrying fluid with it(see figure 9 (e)). This motion
generates vortices, and the dipole jet at the end phase of the maneuver,
providing lift and drag force suitable for a smooth touch-down. As this motion
induces a slower dipole jet, it does not displace the vortex pair further
apart upon impact with the ground, reducing its influence on the rapid drop-
off of the lift and drag force compared to $t^{*}_{os}$ = 0 scenario.

We correlated the induced dipole jet with increased added mass during the
frontal area expansion. The one-dimensional added mass force for an expanding
body can be rewritten as:

$F_{AM}=-\frac{\partial}{\partial t}(m_{a}U)=-\dot{m}_{a}U-m_{a}\dot{U}$ (2)

From this equation, for expanding body, the addition of added mass from
frontal area change creates drag from $-\dot{m}_{a}U$. However, this frontal
area expansion also increases total added mass, making the body difficult to
stop at the latter stage of the maneuver due to the generation of the net
thrust through $-m_{a}\dot{U}$.

In asynchronous motion, $t^{*}_{os}$ = 0 case induces a larger, stronger
dipole jet, increasing the added mass early in the maneuver. This increased
added mass generates higher parasitic thrust through $-m_{a}\dot{U}$ later in
the maneuver, which the bird can use to accelerate after catching the prey.
Conversely, when the frontal area expansion occurs at the low velocity (i.e.,
$t^{*}_{os}$ = 0.5), a slower dipole jet is induced, leading to a smaller
increment in the added mass and a lower value of parasitic thrust, resulting
in smooth touch down during landing or perching.

In summary, this study demonstrates the importance of dipole jets in achieving
specific flying objectives. We observed that by directing the dipole jet in
different directions, birds could generate lift, drag, or thrust force to suit
their needs during landing, perching, or hunting. These findings provide new
insight into the intricate relationship between the dipole jet and the added
mass forces, offering a new perspective on the performance of the bird’s
flight and the design for safer flying vehicles.

Figure 10: (a) Lift and (b) drag coefficient on a finite rectangular wing at
the non-dimensional ground height of $h^{*}$ = 1.5 as a function of $\Xi$.
Scaling of the (c) lift and (d) drag coefficients for all the $\Xi$ and ground
heights considered in the experiment. $C_{L_{avg}}$ is multiplied by a factor
of a, which is the ratio of the pitch time period to the deceleration time
period.

###  3.4 Scaling Laws

The total lift and drag forces generated during the rapid pitching motion are
crucial for achieving specific flying objectives. We analyze time-averaged
lift and drag coefficients across various rapid pitching scenarios to evaluate
flight success. We developed a new scaling law to describe the performance of
the rapidly pitching plates in decelerating motion near the ground.

In synchronous motion at a non-dimensional ground height of $h^{*}=1.5$, we
observed a linear increase in the time-averaged forces with increasing $\Xi$
(for reference, see figure 10 (a) & (b) represented by the symbol,  $\CIRCLE$
). This finding is consistent with a previous study by Polet et al. (2015).
However, in asynchronous motion, we found that executing pitch-up motion at
different stages of deceleration generated a more complex relation between
time-averaged forces and $\Xi$ (represented by the symbol,  $\blacklozenge$ in
figure 10 (a) & (b)). Specifically, we found that increasing the starting time
offsets, $t^{*}_{os}$, from 0 to 0.5, at the same $\Xi$, led to a decrease in
time-averaged lift and drag forces.

To account for this complex relationship, we introduced new scaling relations
by considering the rate of change of maximum lift and drag coefficient,
$\frac{dC_{L_{max}}}{dt}$ & $\frac{dC_{D_{max}}}{dt}$, which were found to be
inversely proportional to the starting time offsets, $t^{*}_{os}$. We
multiplied this rate of change with the $\Xi$ and the averaged lift
coefficient ($C_{L_{avg}}$) by a factor of a, which is the ratio of pitch to
deceleration time period. We used this scaling law to scale the time-averaged
lift and drag coefficient for all tested scenarios, and our experimental data
demonstrate good agreement with this new scaling law (see figure 10 (c) &
(d)).

The main improvement of this scaling law is the connection between the
execution of the pitch-up motion during decelerating motion and the generation
of total lift and drag forces. Our findings indicate that the timing of pitch-
up motion influences the evolution of the flow field around the plates,
leading to changes in the development of lift and drag forces. This scaling
law can be applied to all pitching plates in decelerating motion, regardless
of the ground height, as shown in figure 10 (c) & (d). This modification
simplifies the prediction of unsteady forces on rapidly pitching plates in
decelerating motion, enabling accurate prediction across a broad range of
parameter space.

###  3.5 Modeling the variation of unsteady forces during a perching maneuver

The perching wing experiences two main types of forces: added mass force and
circulatory force.

Added mass force, also known as non-circulatory force, arises due to the
acceleration of the fluid during the unsteady motion of the wing. For a
rapidly pitching plate in decelerating motion, the added mass force
coefficient can be determined using the following equation:

$C_{F_{nc}}=\frac{\pi
c}{2U^{2}_{\infty}}[\dot{\alpha}cos(\alpha)U+sin(\alpha)\dot{U}+c\ddot{\alpha}(1/2-x^{*}_{p})]$
(3)

The first term in this equation denotes the rate of change of added mass,
while the second and third term corresponds to the added mass due to the
accelerating or deceleration of the wing and rotational acceleration,
$\ddot{\alpha}$, respectively. Here, the pivot location on the wing is at the
mid-chord, resulting in the relative distance of $(1/2-x^{*}_{p})$ equal to
zero. When a body is close to the ground, the acceleration of the fluid
between the wing and the ground increases leading to a substantial increase in
the added mass force (Brennen, 1982). To account for this effect, we modified
the added mass equation as:

$C_{F_{nc}}=\frac{\pi
c}{2U^{2}_{\infty}}[\dot{\alpha}cos(\alpha)U+sin(\alpha)\dot{U}+c\ddot{\alpha}(1/2-x^{*}_{p})][1+k(1/2h^{*})^{2}]$
(4)

where k is the constant determined experimentally, with k = 0.002.

In this study, we developed an analytical model to calculate the unsteady
forces acting on a finite wing as it rapidly pitches and decelerates near the
ground. We incorporated the finite wing and unsteady effect by using a
combination of Wagner’s theory and the unsteady lifting line model (LLT)
following Boutet & Dimitriadis (2018). Here, we modified the LLT to include
the ground effect behavior using image vortices.

To represent the flat plate and the trailing vortex sheet, we used the lifting
line approach. From Prandtl’s lifting line theory, the downwash velocity,
$w_{y}$, induced at spanwise location $y$ is given by:

$w_{y}=\frac{-1}{4\pi}\int_{-S/2}^{S/2}\frac{\frac{d\Gamma}{dy_{0}}}{y-y_{0}}{dy_{0}},$
(5)

Here, $\Gamma$ is the strength of the vortex. For the unsteady case involving
a finite wing, $\Gamma$ is a function of time ($t$) and span location ($y$)
distribution and can be represented by a Fourier series with a time varying
Fourier coefficients, $a_{n}$, as:

$\Gamma(t,y)=\frac{1}{2}a_{0}c_{0}U\sum_{n=1}^{N}a_{n}(t)\sin(n\theta),$ (6)

where $a_{0}$ is the lift curve slope, with $a_{0}=2\pi$ for an ideal airfoil,
$c_{0}$ is the chord length of the wing, and $N$ is the number of spanwise
stripes. Here the coordinate transformation of $y=\frac{s}{2}cos(\theta)$ is
applied to map the angle, $\theta$, to the semi-span, $\frac{s}{2}$, position
of the wing. Substituting Eq.(6) into Eq.(5) and applying the Glauert integral
(Glauert, 1983) leads to a simplified form of downwash velocity as:

$w_{y}(t)=-\frac{a_{0}c_{0}U}{4S}\sum_{n=1}^{N}na_{n}(t)\frac{\sin(n\theta)}{sin(\theta)},$
(7)

To include the ground effect, we incorporate an image vortex system that
satisfies the zero normal flow boundary condition on the ground. This image
vortex system induces an upwash on the finite wing, effectively modifying the
tip vortex-induced downwash velocity. The upwash velocity, $w_{I_{y}}$,
induced by the image lifting line can be represented following (Ariyur, 2005):

$w_{I_{y}}(t)=\frac{cos(2\beta)}{4\pi}\int_{-S/2}^{S/2}\frac{(y-y_{0})\frac{d\Gamma(t)}{dy_{0}}\quad{dy_{0}}}{[(y-y_{0})^{2}+4h^{2}cos^{2}(\theta)]},$
(8)

This upwash velocity is the main contribution of the ground effect, which
alters the lift forces on the plate. After coordinate transformation and
substitution of the derivative of $\Gamma$ into Eq. (8), the upwash velocity
can be expressed as:

$w_{I_{y}}(t)=\frac{cos(2\beta)}{\pi}\\\
\int_{0}^{\pi}\frac{(cos(\theta_{0})-cos(\theta))\sum_{n=1}^{m}na_{n}cos(n\theta_{0})}{[(cos(\theta_{0})-cos(\theta))^{2}+16(\frac{h}{b})^{2}cos^{2}(\beta)]}{d\theta_{0}},$
(9)

The unsteady sectional lift coefficient, $c_{l}^{c}(t)$, can be obtained in
terms of $a_{n}(t)$ using the unsteady Kutta-Joukowski theorem (Katz &
Plotkin, 2001):

$c_{l}^{c}(t)=\frac{2\Gamma}{Uc(y)}+\frac{2\dot{\Gamma}}{U^{2}},$ (10)

By substituting the Fourier series representation of the vortex strength
(Eq.6) for $\Gamma$, $c_{l}^{c}(t)$ can be expressed as:

$c_{l}^{c}(t)=a_{0}\sum_{n=1}^{N}(\frac{c_{0}}{c}a_{n}+\frac{c_{0}}{U}{\dot{a}_{n}})\sin(n\theta),$
(11)

The unsteady motion of the flat plate causes a step change in the downwash
velocity, $\Delta w(y)$, along the span. The variation of the circulatory lift
coefficient due to this step change is given in terms of indicial function:

$c_{l}^{c}(t)=a_{0}(y)\Phi(t)\frac{\Delta w(y)}{U},$ (12)

Here, $\Phi(t)$ represents the Wagner function, which can be expressed as:

$\Phi(t)=1-\Psi_{1}e^{-}\frac{\epsilon_{1}U}{b}t-\Psi_{2}e^{-}\frac{\epsilon_{2}U}{b}t,$
(13)

where the constants $\Psi_{1}$ = 0.165, $\Psi_{2}$ = 0.335, $\epsilon_{1}$ =
0.0455 and $\epsilon_{2}$ = 0.3 are derived from Jone’s approximation of the
Wagner function (Jones, 1938).

To capture the continuous lift response of an airfoil that undergoes arbitrary
motion, we use Duhamel’s integral. This method involves superimposing the step
response of the Wagner function $\Phi(t)$ with the differential variation of
the downwash velocity, $w(t,y)$. However, in our case, the flat plate
experiences gradual deceleration, and the free-stream velocity decreases with
time. To account for this variation, we modify Duhamel’s integral formulation
by considering $U=U(t)$, as suggested by Van der Wall (1992) and Hansen et al.
(2004). Based on this, we rewrite Duhamel’s integral formulation with the
time-varying free stream velocity as:

$c_{l}^{c}(s)=\frac{a_{0}}{U}\bigg{(}w(0)\Phi(s)+\int_{0}^{s}\frac{(\partial
w(\tau)}{\partial\tau}\Phi(s-\tau)d\tau\bigg{)},$ (14)

where $s$ is the non-dimensional time scale, which is calculated as
$s=\frac{2}{c}\int_{0}^{t}Udt$.

Applying integration by parts to Duhamel’s integral and substituting the first
order differential equation derived by differentiating the two time lags terms
of the $\Phi(t)$ with respect to $t$, yields the following expression of the
sectional circulatory lift coefficient along the wing span:

$c_{l}^{c}(t,y)=\frac{a_{0}(y)}{U}(w(t,y)(1-\Psi_{1}-\Psi_{2})+y_{1}(t)+y_{2}(t)$
(15)

where the state variables $y_{i}$ are defined as:

$y_{i}(t)=\Psi_{i}\epsilon_{i}\frac{2}{c}\int_{0}^{t}w(t^{\prime},y)U(t^{\prime})e^{-\epsilon_{i}\frac{2}{c}\int_{t^{\prime}}^{t}U(\tau)d\tau}dt^{\prime}$
(16)

In this study, the downwash on the wing is caused by both the motion of the
wing and the 3D wake. The motion of the wing contributes to the downwash
through pitch and angle of attack. The 3D downwash is calculated using
modified lifting line theory, which considers both downwash from the trailing
vortex sheet and upwash from the image vortex sheet. The total downwash
$w(t,y)$ on the wing is expressed as:

${\color[rgb]{1,0,0}w(t,y)=U\alpha_{y}(t)+\dot{\alpha}_{y}(t)d+w_{y}(t)+w_{I_{y}}(t)}$
(17)

where d represents Theodersen’s nondimensional distance.

To remove the integrals from the state variables, new state variables,
$z_{k}$, are introduced:

$z_{k}(t,y)=\int_{0}^{t}e^{-\epsilon_{i}\frac{2}{c}\int_{t^{\prime}}^{t}U(\tau)d\tau}v_{k}(t^{\prime},y)dt^{\prime},\quad
i=1,2$ (18)

where k = 1,2,…6, $v_{1,2}$ = $\alpha$, $v_{3,4}$ = $\frac{w_{y}}{U}$ and
$v_{5,6}$ = $\frac{w_{I_{y}}}{U}$. Here, i = 1 is used for k = 1, 3, and 5,
and i = 2 for k = 2, 4, and 6.

To express the first-order differential equation of Eq. 18, we used Leibniz’s
integral rule:

$\dot{z}_{k}(t,y)=v_{k}-\frac{\epsilon_{i}U}{b}z_{k}(t,y),\quad i=1,2$ (19)

After combining Eqs.(11), (15) and (17), and performing the substitution of
$z_{k}$ into $y_{i}$, we can now turn Eq. (15) at the $j_{th}$ stripe into a
Wagner lifting line matrix equation as:

$\displaystyle\boldsymbol{D_{yj}}\boldsymbol{\dot{a}_{n}}$
$\displaystyle=\boldsymbol{J_{j}}\dot{\textbf{p}}+\boldsymbol{K_{j}}\textbf{p}+\boldsymbol{L_{j}}\textbf{z}+(r(y)(\boldsymbol{W_{yj}}+\boldsymbol{W_{Iyj}})-\boldsymbol{A_{yj}})\boldsymbol{a_{n}}$
(20) $\displaystyle\dot{\textbf{z}}$
$\displaystyle=\boldsymbol{E_{j}}\textbf{z}+\boldsymbol{F_{j}}\textbf{p}+\frac{\textbf{G}}{U}(\boldsymbol{W_{yj}}+\boldsymbol{W_{I_{y}j}})\boldsymbol{a_{n}}$

where,

$\displaystyle\textbf{F}_{j}$ $\displaystyle=[1\ 1\ 0\ 0\ 0\ 0]^{T},$
$\displaystyle\textbf{D}_{yj}$
$\displaystyle=a_{0}\sum_{n=1}^{N}\frac{c_{0}}{U}\sin(n\theta_{j}),$
$\displaystyle\textbf{A}_{yj}$
$\displaystyle=a_{0}\sum_{n=1}^{N}\frac{c_{0}}{c}\sin(n\theta_{j}),$
$\displaystyle\textbf{E}_{j}$
$\displaystyle=\frac{-U}{b}diag(\epsilon_{i=1},\quad\epsilon_{i=2},...),$ z
$\displaystyle=[z_{k=1}\quad z_{k=2}\quad...]^{T},$ G $\displaystyle=[0\ 0\ 1\
1\ 0\ 0;\ 0\ 0\ 0\ 0\ 1\ 1]^{T},$ p $\displaystyle=[\alpha(t,y)],$
$\displaystyle\textbf{J}_{j}$ $\displaystyle=\frac{a_{0}(y)}{U}\Phi(0)d,$
$\displaystyle\textbf{K}_{j}$
$\displaystyle=\frac{a_{0}(y)}{U}[U\Phi(0)+d\dot{\Phi}(0)],$
$\displaystyle\textbf{L}_{j}$
$\displaystyle=\frac{a_{0}(y)U}{b}[\Psi_{1}\epsilon_{1}(1-\epsilon_{1}\frac{d}{b})\quad\Psi_{2}\epsilon_{2}(1-\epsilon_{2}\frac{d}{b})\quad\Psi_{1}\epsilon_{1}\quad\Psi_{2}\epsilon_{2}\quad\Psi_{1}\epsilon_{1}\quad\Psi_{2}\epsilon_{2}]^{T},and$
$\displaystyle r(y)$ $\displaystyle=\frac{a_{0}(y)\Phi(0)}{U}$ .

In this study, the wing is divided into $N$ spanwise sections. When we apply
Eq.(20) to all the N sections, a system of 9N differential equations is
formed. A MATLAB-built ODE function, ode15s, is used to solve the given
systems of ODE. After computing the Fourier coefficients, $a_{n}$, Eq.(10) is
used to calculate the lift coefficient along the wing span.

Figure 11: Comparison of lift coefficients on the flat plate: (a) Experimental
and predicted results for three values of $\Xi$ at $h^{*}$ = 1.5; (b)
Experimental ground effect data and predicted results for $\Xi$ = 0.6.

Figure 11 compares the time-varying lift coefficient, $C_{L}$, predicted by
our analytical model with the experimental results. In figure 11 (a), a steep
rise in $C_{L}$ is observed during the initial deceleration phase for all
values of $\Xi$, consistent with the experimental findings. This suggests that
both the non-circulatory and the circulatory forces play a dominant role in
generating unsteady forces in pitching plates during deceleration.
Subsequently, $C_{L}$ decreases, as observed experimentally, due to a
reduction in non-circulatory and circulatory forces. The analytical model
successfully captures the general trend of the instantaneous lift coefficient
evolution observed in the experimental results (Figure 11 (a)). However,
accurate prediction of lift evolution in pitching plates requires a precise
estimation of vortex growth and detachment from the wing surface, which the
present analytical model lacks. This discrepancy explains the disparity
between the predicted and experimental $C_{L}$ values.

Figure 11 (b) compares the ground effect results obtained from experiments
with the predicted results from analytical model for $\Xi$ = 0.6. To
incorporate the ground effect, we modified the added mass equation and
integrated an image vortex system into the circulatory force analysis. With
this modification, the present analytical model successfully predicts the
observed increase in the peak $C_{L}$ as the wing approaches the ground and
captures the corresponding drop in $C_{L}$ and increase in negative lift as
the wing gets closer to the ground. However, there is a slight deviation
between the predicted peak $C_{L}$ and the experimental values for all ground
heights, primarily due to model’s limited ability to accurately estimate
vortex growth and detachment from the wing surface. Nevertheless, the present
model effectively captures the general trend in the evolution of $C_{L}$ and
the influence of ground effect on rapidly pitching plates during deceleration.

##  4 Conclusion

We used a rapid pitch-up motion of a finite flat plate in a decelerating flow
as a simple model to investigate the effect of frontal area change on unsteady
forces and flow physics in perching or hunting birds. Experiments were
conducted to measure the instantaneous force and flow development at different
pitch rates and decelerations. We tested the same kinematics at different
ground heights to explore the unsteady ground effect experienced by the
rapidly pitching birds. We developed a simple analytical model that
incorporates the added mass forces and the circulatory force to predict the
instantaneous lift force on the wing. The predicted results were compared with
the experimental data to validate the accuracy of the model.

To safely land or perch, birds need enough lift force to support their weight
and drag force to decelerate to a complete stop. Our research shows that in
synchronous motion increasing the shape change number $\Xi$ leads to higher
lift and drag forces on the wing. This higher unsteady force is the outcome of
the combined increase in non-circulator and circulatory force resulting from
the higher pitch rate. While this higher drag force is crucial for the rapid
dissipation of the kinetic energy during landing or perching, however, this
higher lift force may prevent the bird from landing at the intended location
or altitude.

In asynchronous motion, varying the starting time offsets between the
deceleration and pitch-up motion results in a distinct evolution of
instantaneous lift and drag forces. For the starting time offset, $t^{*}_{os}$
= 0, the plate generates a high initial lift and drag force but experiences a
rapid drop-off of the aerodynamic forces later in the motion. However,
delaying the rapid pitch-up motion until late in the deceleration (i.e.,
$t^{*}_{os}$ = 0.5) allows the wing to generate lift and drag at the end of
the motion. In this case, the wing generated higher positive drag force and
near-zero lift force, which is suitable for landing on the initial landing
location without gaining altitude. Through flow field analysis, we show that
when a rapidly pitching plate is synchronized with the start of the
decelerating motion (i.e., $t^{*}_{os}$ = 0 case), the plate can transfer a
higher added mass energy to the fluid by generating a stronger dipole jet
during the initial stage of the unsteady motion. Subsequently, the wing
recovers this energy as parasitic thrust, which can be utilized by hunting
birds for acceleration after capturing prey. It demonstrates that birds
utilize rapid wing pitching during deceleration to accomplish diverse flying
objectives, which can be achieved either by recovering added mass energy later
in the maneuver or delaying the formation of vortices towards the end of the
motion.

Our experiments revealed that the ground effect can significantly enhance the
aerodynamic performance of rapidly pitching plates in deceleration motion.
Regarding ground heights $(0.04<h^{*}<1.5)$, our results showed that the
ground proximity affects the initial rise in the lift force, which increased
as the wing gets closer to the ground. However, the initial drag force
remained independent of ground height. The ground effect impacts the parasitic
thrust, which increased for all $\Xi$ values when the wing is close to the
ground. Although minor disparities exist between the predicted and
experimental results, our present analytical model effectively captures the
general trend in the evolution of instantaneous $C_{L}$ on rapidly pitching
plates during deceleration. Notably, the model predicts an increase in the
$C_{L}$ as the wing approaches the ground, accurately reflecting the influence
of ground proximity and demonstrating strong agreement with the experimental
results.

Overall this study highlights the significance of tuning rapid pitching with
deceleration, as observed in perching and hunting birds, which employ
strategies such as the recovery of added mass energy for acceleration or the
delayed formation of vortices for smooth landing or perching. This improved
understanding of the highly unsteady performance of natural flyers contributes
to the design of safer and more efficient flying vehicles.

## Acknowledgments

Acknowledgments

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