2. Experimental approach

For experimental ease, our study is conducted in a water tank 140 cm tall, 100 cm wide and 10.5 cm deep (figure 1 a). Our wings correspond to flat rectangular plates of chord length $l$ , thickness $h$ and span length $L$ and are released at the top of the tank and freely settle through water of density ${\it\rho}$ under the influence of gravity $\boldsymbol{g}$ , see figure 1(a). Wings are characterized by their density ${\it\rho}_{s}$ and bending rigidity in the chordwise direction ${\it\kappa}$ . In order to both settle to the bottom of the tank and elastically deform, the wings are required to be both negatively buoyant and flexible. Consequently, we manufacture wings from two different materials. We use metallic brass of density $8500~\text{kg}~\text{m}^{-3}$ to allow the wings to settle in water as well as the elastic polymer vinylpolysiloxane of density $1050~\text{kg}~\text{m}^{-3}$ to allow the wings to be compliant. The wings are assembled by arranging equally spaced brass bars in a custom-made mould before casting the elastic polymer (figure 1 a). The spanwise dimension of the wings, $L=10$  cm, closely coincides with the depth of the tank in order to constrain the experiment to be two dimensional. Millimetre-sized polished stainless steel beads are fixed at the four corners of the wing, allowing the wing to freely move in water with negligible friction against the glass wall of the tank.

The dynamics of the passive wings is characterized by two non-dimensional numbers,

(2.1a,b ) $$\begin{eqnarray}I^{\star }=\frac{{\it\rho}_{s}h}{{\it\rho}l}\quad \text{and}\quad C_{y}=\frac{{\it\rho}U^{2}l^{3}}{{\it\kappa}},\end{eqnarray}$$

where $U=\sqrt{({\it\rho}_{s}-{\it\rho})hg/{\it\rho}}$ is the characteristic settling velocity deduced from a force balance between weight and buoyancy on the one hand and hydrodynamics on the other. Here, $I^{\star }$ is the non-dimensional moment of inertia, which can also be interpreted as the inverse of the added mass parameter. In our study, the material densities are not varied and $I^{\star }$ is a geometric parameter characterizing the cross-sectional aspect ratio $h/l$ of the wings. The non-dimensional number $C_{y}$ is the Cauchy number, which corresponds to the ratio of hydrodynamic to elastic bending forces and increases with the flexibility of the wing. In all experiments, the Reynolds number is of the order of $\mathit{Re}\approx 10^{5}$ and does not vary significantly between experiments. In this study, the influences of $I^{\star }$ and $C_{y}$ were investigated by varying the cross-sectional geometry of the wing and the Young’s modulus of the elastic polymer independently. We considered five wing cross-sectional geometries, $I^{\star }=0.05$ , $0.07$ , $0.10$ , $0.17$ , $0.24$ , by varying the thickness of the brass bars ( $h=0.4{-}1.6~\text{mm}$ ) and the chord length of the wings ( $l=41{-}80$  mm). All five cross-sectional geometries have low thickness to chord aspect ratio, $h/l=0.005{-}0.039$ , and the wings can be considered to be thin flat plates. All of these wings undergo a fluttering motion in agreement with previous experimental studies, which have determined the transition to tumbling motion to occur beyond a critical value of the non-dimensional moment of inertia of $I_{crit}^{\star }\approx 0.4$ (Belmonte et al. Reference Belmonte, Eisenberg and Moses1998). Using nine different elastic polymers with Young’s moduli ranging from $E=0.26$ to 1.2 MPa, the bending rigidity in the chordwise direction ${\it\kappa}$ is varied. The natural frequencies of the flexible wings are no less than 10 Hz. Fully rigid wings are also considered by fixing thin mylar films to the upper and lower surfaces of the wings. The Cauchy number ranges between 0 for rigid wings and 70 for the most flexible ones. The range of Reynolds and Cauchy numbers investigated in our experiments is relevant to many problems of objects, paper, leaves and seedpods falling in air.

Wings are released in the water tank and their descent is recorded using high-speed imaging at 150–250 f.p.s. Figure 1(b) represents successive snapshots recorded with our experimental set-up. The videos are processed to extract the two-dimensional trajectory of the wing $\boldsymbol{X}(t)=(x(t),y(t))$ , the wing velocity $U(t)=(v_{x}(t),v_{y}(t))$ and the pitch angle ${\it\theta}(t)$ between the horizontal $\hat{x}$ and chord $\hat{x}^{\prime }$ directions, figure 1(a). The side-to-side fluttering motion is characterized by turning points, where the wings stop and change direction, separated by gliding intervals, where the wings descend due to gravity. Characteristic quantities are deduced from the analysis of the trajectories: the horizontal ${\rm\Delta}x$ and the vertical ${\rm\Delta}y$ distances between consecutive turning points, the frequency of the oscillatory motion $f$ , the average vertical descent velocity of the wing $U_{d}$ and the characteristic flight time ${\it\tau}_{f}=l/U_{d}$ required to descend over one chord length $l$ at the average descent rate $U_{d}$ (figure 1 b). Wing shape is characterized by the deflection, which corresponds to the distance from a point along the deformed camberline to the straight chordline joining the leading and trailing edges. The extent of elastic deformation is recorded by measuring the maximum deflection along the wing ${\it\delta}$ . In addition, the two-dimensional flow velocity field $\boldsymbol{u}$ in the tank is measured in separate sets of experiments using double frame particle image velocimetry (PIV). Figure 1(c) represents the typical vorticity field left behind a fluttering wing. All experimental data are non-dimensionalized with the characteristic velocity as a reference velocity $U_{ref}=U$ , the dimension of the chord as a reference length $l_{ref}=l$ and the reference time scale $t_{ref}=l/U$ . Henceforth, all variables are non-dimensional unless otherwise specified.

Figure 1.

(a) Experimental set-up. (b) Superimposed snapshots of the fluttering motion ( $C_{y}=0$ and $I^{\star }=0.1$ ). Time between snapshots: $125$  ms. (c) Particle image velocimetry measurements of the vorticity and velocity fields ( $C_{y}=10$ and $I^{\star }=0.1$ ). The wing is represented by a line and the wing circulation ${\rm\Gamma}$ and wake vortices ${\rm\Gamma}_{wake}$ are highlighted by coloured arrows.

3. Experimental results

We proceed by detailing our experimental observations. For each cross-sectional geometry characterized by $I^{\star }$ , the effect of flexibility is isolated by studying wings with increasing Cauchy number from values corresponding to rigid wings, $C_{y}=0$ , to highly flexible wings, $C_{y}\approx 70$ . Following their release, the wings settle in a stable periodic two-dimensional left-to-right fluttering motion (see the supplementary movies available at http://dx.doi.org/10.1017/jfm.2015.1). The fluttering motion is always observed for all but the most flexible wings. For very flexible wings ( $C_{y}\geqslant 60$ ), the stability of the fluttering motion is lost to an unstable flapping motion (Shelley et al. Reference Shelley, Vandenberghe and Zhang2005), resulting in rapid descent, see the supplementary movies. The flexibility at which this transition occurs is in agreement with previous experimental work (Shelley et al. Reference Shelley, Vandenberghe and Zhang2005). We henceforth restrict our attention to the stable fluttering motion, which is the only dynamics observed for $C_{y}\leqslant 60$ .

Flexible wings deform during the fluttering motion. Typical wing shapes are reproduced in figure 2(a). Rigid wings, $C_{y}=0$ , remain flat throughout their descent, while flexible wings exhibit small upward bending deformation ${\it\delta}$ , generally not exceeding $5\,\%$ of the chord length $l$ . Figure 2(c) represents the time variations of the deformation ${\it\delta}$ during the fluttering motion for $I^{\star }=0.1$ and $C_{y}=15$ . During gliding intervals, the deformation does not exceed $1\,\%$ and the wings can be considered to remain flat. Elastic deformations are restricted to short time periods at the turning points, where ${\it\delta}$ reaches a maximal value $\bar{{\it\delta}}$ of the order of $4\,\%$ , see figure 2(c). In our experiments, the chord length does not exceed $8$  cm and the dimensional deflection does not exceed $5$  mm. In figure 2(a), the white arrows indicate the frames in the sequence of snapshots, when the maximal deformation $\bar{{\it\delta}}$ is observed. The wing deformations $\bar{{\it\delta}}$ remain very small for all wings, as can also be seen in the supplementary movies. Figure 2(d) represents the maximal wing deformation $\bar{{\it\delta}}$ recorded at the turning points for wings of increasing flexibility. The elastic deformation $\bar{{\it\delta}}$ is found to increase linearly with the flexibility of the wing $C_{y}$ .

Figure 2.

(a) Superimposed snapshots of the fluttering motion for $I^{\star }=0.1$ and increasing $C_{y}$ . White arrows highlight wing shapes, when the deformation ${\it\delta}$ reaches its maximum value $\bar{{\it\delta}}$ . (b) Circulation measurements deduced from high-speed imaging as a function of time for a fluttering wing ( $I^{\star }=0.1$ and $C_{y}=15$ ) (c). Bending deformation ${\it\delta}$ as a function of time for the same fluttering wing as in (b). (d) Maximum bending deformation $\bar{{\it\delta}}$ measured at the turning point as a function of the Cauchy number $C_{y}$ for fluttering wings ( $I^{\star }=0.1$ ).

Figure 3.

Influence of wing flexibility on flight characteristics. For each $I^{\star }$ , the wing flexibility $C_{y}$ is varied and the dependence of the flight characteristics with respect to the Cauchy number $C_{y}$ is recorded. The panels correspond to (a)  ${\rm\Delta}x$ , the horizontal amplitude of the fluttering motion, (b)  ${\rm\Delta}y$ , the vertical amplitude of the fluttering motion, (c)  $f$ , the frequency of the fluttering motion, (d)  ${\it\tau}_{f}$ , the characteristic flight time. All quantities are non-dimensional.

While the extent of elastic deformation remains small, its effect on the flight characteristics is striking. For each value of $I^{\star }$ , the fluttering amplitudes ${\rm\Delta}x$ and ${\rm\Delta}y$ decrease severalfold as $C_{y}$ is increased, see figure 3(a,b). Concurrently, the frequency $f$ of the fluttering motion increases severalfold, see figure 3(c). The characteristic flight time ${\it\tau}_{f}$ is the most relevant quantity in our study as it characterizes how long the wing remains airborne. The non-dimensional flight time ${\it\tau}_{f}$ is precisely the inverse of the non-dimensional average descent velocity $U_{d}$ . The flight time ${\it\tau}_{f}$ strongly depends on the wing flexibility $C_{y}$ (figure 3 d). For each wing geometry $I^{\star }$ , ${\it\tau}_{f}$ increases from its value for rigid wings at $C_{y}=0$ to a maximum at some optimal flexibility $C_{y}^{\star }$ . For $C_{y}\geqslant C_{y}^{\star }$ , ${\it\tau}_{f}$ decreases with increasing $C_{y}$ (figure 3 d). For $I^{\star }=0.05$ , the flight time of wings of optimal flexibility is $1.22$ times longer than for rigid wings. This increased flight time is enhanced at higher $I^{\star }$ values. For the highest value, $I^{\star }=0.24$ , the optimally flexible wing remains aloft $2.5$ times longer than its rigid counterpart, corresponding to a more than twofold increase in flight time. While the fluttering frequency does increase with flexibility, the dimensional values of $f$ remain of the order of $1$  Hz, which is one order of magnitude lower than the natural frequencies of the wings. The optimal flight time due to flexibility is hence not caused by a resonant mechanism. The effect of flexibility on the flight characteristics is best observed by watching recorded videos of fluttering wings of identical geometry and increasing flexibility provided in the supplementary material.

Velocity and vorticity fields obtained from PIV measurements provide key insight into the fluttering dynamics and shed light on the mechanisms generating wing circulation. Figure 1(c) represents the typical vorticity field produced by a descending wing. Particle image velocimetry measurements were performed for wings of varying geometry $I^{\star }$ and flexibility $C_{y}$ . Two key vortical elements can be identified in the vorticity distribution. First, the bound vorticity on the wing corresponds to the wing circulation at the origin of the Joukowski lift. Second, vortices are shed at each turning point: a strong vortex and an associated much weaker counter-rotating one (see figure 1 c). Outside of these two vortical elements, little vorticity is present in the flow. This characteristic vortex distribution is observed for all fluttering wings of varying $C_{y}$ and $I^{\star }$ . We estimate the wing circulation during the gliding interval ${\rm\Gamma}$ and the circulation around the asymmetric vortex pair shed at the turning point ${\rm\Gamma}_{wake}$ . We do so by evaluating the line integral ${\rm\Gamma}=\oint _{\mathscr{C}}\boldsymbol{u}\boldsymbol{\cdot }\text{d}\boldsymbol{l}$ , where $\mathscr{C}$ represents a closed contour closely following the vortical element of interest and $\boldsymbol{u}$ is the flow velocity interpolated on $\mathscr{C}$ . In addition, the wing circulation ${\rm\Gamma}$ was also measured from the high-speed imaging experiments. The acceleration of the wing is computed from the recorded time-resolved wing trajectory $\boldsymbol{X}(t)=(x(t),y(t))$ . One can directly deduce the total aerodynamic force acting on the wing. Assuming a Joukowski lift ${\it\rho}{\rm\Gamma}|\boldsymbol{U}|$ (Andersen et al. Reference Andersen, Pesavento and Wang2005), one can estimate the wing circulation from the lift force. Figure 2(b) represents the time variations of the circulation deduced from high-speed imaging for $I^{\star }=0.1$ and $C_{y}=15$ . During gliding intervals, the circulation remains relatively constant at a value of $0.008~\text{m}^{2}~\text{s}^{-1}$ . Turning points correspond to the shaded areas, for which the estimate is not expected to be accurate as the wing velocity $\boldsymbol{U}$ vanishes. The wing circulation was estimated separately using PIV and high-speed imaging and the measurements were in perfect agreement and confirmed the accuracy of the PIV measurements. For a given cross-sectional geometry $I^{\star }$ , we observe flexible wings to develop larger wing circulation ${\rm\Gamma}$ than their rigid counterparts and to shed stronger vortices ${\rm\Gamma}_{wake}$ at the turning points (see figure 4). The increase in wing circulation due to flexibility is significant. For $I^{\star }=0.24$ and $C_{y}=11$ , the circulation is $2.1$ times larger than for the corresponding rigid wing. The circulation increases deduced from the PIV measurements and from high-speed imaging are in agreement.

Figure 4.

Dependence of ${\rm\Delta}{\rm\Gamma}_{e}={\rm\Gamma}-{\rm\Gamma}_{0}$ on wing flexibility $C_{y}$ . Each data set corresponds to a fixed cross-sectional geometry with $0.05\leqslant I^{\star }\leqslant 0.24$ . Inset: the circulation for rigid wings ( $C_{y}=0$ ) is independent of wing geometry $I^{\star }$ .

4. Theoretical scalings

In the following, we elucidate the physical mechanisms responsible for the increase in lift and flight time. Hence, we focus exclusively on flexibilities between the rigid limit $C_{y}=0$ and the optimal value $C_{y}^{\star }$ , for which the flight time is maximum. In this discussion, we consider a simplified vorticity distribution as follows. The bound vorticity on the wing is reduced to a single point vortex ${\rm\Gamma}$ attached to the wing and each asymmetric vortex pair is represented by a point vortex at the turning point, whose circulation ${\rm\Gamma}_{wake}$ corresponds to the circulation around the vortex pair (figure 1 c). Outside of these vortices, we assume the flow to be irrotational in agreement with our PIV data. This simplified vorticity distribution is characteristic of the observed distribution for all fluttering wings.

During a gliding interval, the wing descends from one turning point to the following one under the influence of gravity with a circulation of $+{\rm\Gamma}$ . At the turning point, the wing comes to rest and sheds a wake vortex $+{\rm\Gamma}_{wake}$ before resuming flight in the opposite direction with a circulation $-{\rm\Gamma}$ . In this simplified vorticity field, Kelvin’s circulation theorem has two important consequences: (i) ${\rm\Gamma}$ and ${\rm\Gamma}_{wake}$ remain constant in time and (ii) ${\rm\Gamma}_{wake}=2{\rm\Gamma}$ . This equality follows directly from the conservation of circulation for a contour around a turning point, considering that the circulation of the wing is initially $+{\rm\Gamma}$ and becomes $-{\rm\Gamma}$ after the wake vortex ${\rm\Gamma}_{wake}$ is shed. This problem is not without correspondence with classical starting vortices. These two inferences are verified experimentally from the measurements of circulation around the wing and around wake vortices from each PIV data set. For each value of $I^{\star }$ , ${\rm\Gamma}$ varied by no more than $8\,\%$ during the gliding interval and the ratio of wake to wing circulation was ${\rm\Gamma}_{wake}/{\rm\Gamma}=2.01\pm 0.08$ . We conclude that the wing circulation ${\rm\Gamma}$ is created predominantly at the turning points and remains relatively constant through the subsequent gliding interval.

The following physical picture emerges. Wing flexibility acts at the turning points, where elastic deformations occur. These passive deformations allow flexible wings to shed stronger wake vortices and to generate larger wing circulation than their rigid counterparts of equal $I^{\star }$ . During gliding intervals, both rigid and flexible wings experience lift forces, which depend on the wing circulation created at the turning point. We proceed by deriving theoretical scalings to rationalize the effect of flexibility on fluttering wings.

4.1. Scaling for the wing circulation

Generally, circulation is created when wings with sharp edges are accelerated and vorticity is shed at the trailing edge. There is an analogy between the wake vortices in our experiment and classical starting vortices. At the turning point, the wing is first accelerated by gravity and fluid flows around the leading and trailing edges. Vorticity develops at the two wing tips and the pressure distribution corresponding to this vorticity distribution creates a bending moment responsible for the brief upwards bending deflection ${\it\delta}$ observed (see figure 2 c). Thereafter, the vorticity at the trailing edge is shed to satisfy the Kutta condition, cancelling the bending moment, and the trailing edge is further accelerated by elastic forces until the wing returns to a flat configuration. This description agrees with the earlier observation that deformations arise only at the turning points.

Two forces thus contribute to the acceleration experienced by the wing tips: gravity and wing elasticity. Accordingly, the wing circulation is written as the sum of two terms, ${\rm\Gamma}={\rm\Gamma}_{0}+{\rm\Delta}{\rm\Gamma}_{e}$ , where ${\rm\Gamma}_{0}$ is the circulation due to the gravitational acceleration of the equivalent rigid wing and ${\rm\Delta}{\rm\Gamma}_{e}$ characterizes the contribution of elasticity. Circulation scales as the product of a characteristic length times a characteristic velocity. Theoretical scalings for each contribution ${\rm\Gamma}_{0}$ and ${\rm\Delta}{\rm\Gamma}_{e}$ can be derived. Gravity accelerates the wing downwards from rest to a descent velocity, which scales with the characteristic settling speed $U$ . The dimensional circulation due to gravity is hence expected to scale with ${\rm\Gamma}_{0}\sim lU$ , which in non-dimensional terms translates to a constant ${\rm\Gamma}_{0}$ independent of $I^{\star }$ and $C_{y}$ . This scaling is validated experimentally by measuring the circulation around rigid wings, ${\rm\Gamma}(C_{y}=0)={\rm\Gamma}_{0}$ , for which the contribution of elasticity vanishes. The circulation ${\rm\Gamma}_{0}$ is constant and independent of $I^{\star }$ in agreement with the scaling (see inset in figure 4). Next, we consider the additional ${\rm\Delta}{\rm\Gamma}_{e}$ generated when the bending energy due to the deformation ${\it\delta}$ is released. In this case, the relevant length scale is the maximal deformation at the turning point $\bar{{\it\delta}}$ and the relevant velocity scale is the velocity of the wing tip $\bar{{\it\delta}}/{\it\tau}$ due to elastic forces, where ${\it\tau}$ is the characteristic time of the elastic response. The following scaling for the dimensional circulation can be deduced: ${\rm\Delta}{\rm\Gamma}_{e}\sim \bar{{\it\delta}}(\bar{{\it\delta}}/{\it\tau})$ . The non-dimensional deflection scales linearly with the Cauchy number, $\bar{{\it\delta}}/l\sim C_{y}$ , in agreement with our experimental observations of ${\it\delta}$ (see figure 2 d). The non-dimensional time of the elastic response scales with ${\it\tau}/t_{ref}\sim C_{y}^{1/2}$ . Together, these scalings yield for the non-dimensional circulation

(4.1) $$\begin{eqnarray}{\rm\Delta}{\rm\Gamma}_{e}\sim C_{y}^{3/2}.\end{eqnarray}$$

The $3/2$ exponent of this nonlinear scaling implies that wing circulation and wake vortices greatly increase with flexibility. This result is verified against our experimental measurements of ${\rm\Gamma}$ for wings of varying flexibility $C_{y}$ . For each different wing geometry $I^{\star }$ , we consider the experimental data recorded for wings, for which we observed an increase in flight time. These wings are characterized by a Cauchy number ranging from the rigid limit $C_{y}=0$ to the value at which the maximal flight time was recorded $C_{y}=C_{y}^{\star }$ . For all flexible wings, ${\rm\Delta}{\rm\Gamma}_{e}$ is estimated experimentally as ${\rm\Delta}{\rm\Gamma}_{e}={\rm\Gamma}-{\rm\Gamma}_{0}$ , where ${\rm\Gamma}$ is the circulation measured around the flexible wing and ${\rm\Gamma}_{0}$ is the circulation measured around its rigid counterpart of identical geometry. For each separate wing geometry $I^{\star }$ , this $3/2$ exponent is verified experimentally by plotting ${\rm\Delta}{\rm\Gamma}_{e}(C_{y})$ on a ‘log–log’ scale, see figure 4. At the optimal $C_{y}^{\star }$ , ${\rm\Delta}{\rm\Gamma}_{e}$ is of the same order of magnitude as ${\rm\Gamma}_{0}$ and the increase in wing circulation due to flexibility is significant.

4.2. Scalings for the flight characteristics

We now turn to the gliding intervals and quantify the effect of increased circulation on the flight characteristics. Previous studies (Mahadevan Reference Mahadevan1996; Belmonte et al. Reference Belmonte, Eisenberg and Moses1998; Andersen et al. Reference Andersen, Pesavento and Wang2005) have investigated the falling dynamics of wings with reduced models adapted from Kirchhoff’s ordinary differential equations (ODEs) for the motion of bodies immersed in an irrotational flow. The non-dimensional momentum equations can be written as

(4.2a ) $$\begin{eqnarray}\displaystyle I^{\star }\dot{v}_{x^{\prime }}-(I^{\star }+1)\dot{{\it\theta}}v_{y^{\prime }} & = & \displaystyle -{\rm\Gamma}v_{y^{\prime }}-D_{x^{\prime }}-\sin {\it\theta},\end{eqnarray}$$

(4.2b ) $$\begin{eqnarray}\displaystyle (I^{\star }+1)\dot{v}_{y^{\prime }}+I^{\star }\dot{{\it\theta}}v_{x^{\prime }} & = & \displaystyle +{\rm\Gamma}v_{x^{\prime }}-D_{y^{\prime }}-\cos {\it\theta},\end{eqnarray}$$

where inertia and added mass appear on the left-hand side, while the right-hand side includes the lift force derived from the Kutta–Joukowski theorem, the drag force and gravity. Here, $v_{x^{\prime }}$ and $v_{y^{\prime }}$ are the components of the wing velocity $\boldsymbol{U}$ in the frame co-rotating with the wing (see figure 1 a), $D_{x^{\prime }}$ and $D_{y^{\prime }}$ are the components of the drag force.

Previous studies rely crucially on empirical quasi-steady expressions for the wing circulation as a function of the instantaneous wing motion $v_{x^{\prime }}$ , $v_{y^{\prime }}$ and $\dot{{\it\theta}}$ (Belmonte et al. Reference Belmonte, Eisenberg and Moses1998; Andersen et al. Reference Andersen, Pesavento and Wang2005). In contrast, we have deduced from our experiments that the lift force during gliding only depends on the circulation level generated at the turning point and that it remains constant throughout the gliding motion. Since no elastic deformations occur during gliding intervals, the dynamics of rigid and flexible wings can be modelled by (4.2) assuming a constant wing circulation (Mahadevan Reference Mahadevan1996), given by (4.1). Assuming that ${\rm\Gamma}$ remains constant simplifies (4.2) and allows the derivation of theoretical scalings for the flight characteristics. We note that by assuming a constant ${\rm\Gamma}$ , the shedding of the wake vortex ${\rm\Gamma}_{wake}$ and subsequent jump in wing circulation from ${\rm\Gamma}$ to $-{\rm\Gamma}$ at the turning point is not adequately represented. Equations (4.2) hence only model a single gliding interval, corresponding a wing started from rest and whose motion is prescribed by (4.2) with constant  ${\rm\Gamma}$ .

Figure 5.

Comparison between experimental data recorded with high-speed imaging (symbols) and theoretical scalings (dashed line). Each data set corresponds to a given cross-sectional geometry. For each $I^{\star }$ , the wing flexibility $C_{y}$ is varied. The panels correspond to scalings for (a)  ${\rm\Delta}x$ , the horizontal amplitude of the fluttering motion, (b)  ${\rm\Delta}y$ , the vertical amplitude of the fluttering motion, (c)  $f$ , the frequency of the fluttering motion, (d)  ${\it\tau}_{f}$ , the characteristic flight time. Scalings are expressed in terms of the wing circulation ${\rm\Gamma}$ and the added mass parameter $1/I^{\star }$ . Here, ${\rm\Delta}x$ , ${\rm\Delta}y$ , $f$ and ${\it\tau}_{f}$ are non-dimensional.

Equations (4.2) can be written in terms of $v_{x}$ and $v_{y}$ , the horizontal and vertical components of $\boldsymbol{U}$ , and further simplified on the basis of experimental observations to derive theoretical scalings. We assume $I^{\star }\ll 1$ , the pitch angle ${\it\theta}\ll 1$ during the gliding segments, leading to the simplified system

(4.3a ) $$\begin{eqnarray}\displaystyle & \displaystyle I^{\star }\dot{v}_{x}=-{\rm\Gamma}v_{y}-D_{x}, & \displaystyle\end{eqnarray}$$

(4.3b ) $$\begin{eqnarray}\displaystyle & \displaystyle \dot{v}_{y}=+{\rm\Gamma}v_{x}-D_{y}-1. & \displaystyle\end{eqnarray}$$

Considering that the scale of the fluttering dynamics is determined by the inertia, added mass, lift and gravity terms in the coupled ODEs (4.3), we can directly deduce theoretical scalings for the different terms in the equations: the horizontal speed $v_{x}\sim 1/{\rm\Gamma}$ , the vertical descent speed $v_{y}\sim \sqrt{I^{\star }}/{\rm\Gamma}$ and the time scale of the periodic motion $1/f\sim \sqrt{I^{\star }}/{\rm\Gamma}$ . Combination of these scalings yields scalings for the flight characteristics:

(4.4a−d ) $$\begin{eqnarray}{\rm\Delta}x\sim \frac{\sqrt{I^{\star }}}{{\rm\Gamma}^{2}},\quad {\rm\Delta}y\sim \frac{I^{\star }}{{\rm\Gamma}^{2}},\quad f\sim \frac{{\rm\Gamma}}{\sqrt{I^{\star }}},\quad {\it\tau}_{f}\sim \frac{{\rm\Gamma}}{\sqrt{I^{\star }}}.\end{eqnarray}$$

Equations (4.4) predict that the fluttering amplitudes ${\rm\Delta}x$ and ${\rm\Delta}y$ decrease with the wing circulation ${\rm\Gamma}$ squared and the frequency $f$ and flight time ${\it\tau}_{f}$ linearly increase with ${\rm\Gamma}$ . From (4.4), the lift coefficient $C_{l}$ can also be shown to be quadratic in ${\rm\Gamma}$ , $C_{l}\sim {\rm\Gamma}^{2}$ . These scalings are verified experimentally by plotting the non-dimensionalized flight characteristics measured from high-speed imaging as a function of the non-dimensional circulation measured from PIV experiments. Figure 5 presents experimental results on a ‘log–log’ scale for each separate wing geometry  $I^{\star }$ . For all four flight characteristics, the quadratic and linear scalings in ${\rm\Gamma}$ (4.4) are verified for each $I^{\star }$ . The Cauchy number $C_{y}$ does not directly appear in the scalings (4.4). However, as discussed previously, the circulation ${\rm\Gamma}$ depends on the flexibility since ${\rm\Gamma}{-}{\rm\Gamma}_{0}$ directly scales with $C_{y}^{3/2}$ .

The theoretical scalings (4.4) depend on the wing dynamics through the non-dimensional moment of inertia  $I^{\star }$ , see (4.3). Taking into account the dependence on $I^{\star }$ allows for a full collapse of all the different experimental data sets with the theoretical scalings, requiring no fitting parameters. The full data collapse in figure 5 is in stark contrast to the data presented in figure 3, where the flight characteristics are presented as a function of the flexibility $C_{y}$ . This collapse validates the physical picture described above. Combination of the theoretical analyses of the turning points and the gliding intervals fully rationalizes all experimental observations. For example, small passive elastic deformations of wings of $I^{\star }=0.24$ occur at the turning points and lead to a more than twofold increase in ${\rm\Gamma}$ relative to their rigid counterparts, figure 4. This increase in circulation results in a more than twofold increase in flight time and fourfold increase in lift coefficient, see figure 5.