Universal vortex formation time of flapping flight

Significance

Through the action of flapping, biological flyers generate a vortex upon each wing known as the leading-edge vortex (LEV), which has been shown to enhance aerodynamic force production. However, the LEV does not grow indefinitely. Here, by examining the limits of LEV growth and its growth rate, we show that the animal’s wing stroke lasts until the LEV reaches its natural growth limit. These findings suggest that the flapping frequency of insects, birds, and bats is governed by this intrinsic constraint.

Abstract

Biological flyers periodically flap their appendages to generate aerodynamic forces. Extensive studies have made significant progress in explaining the physics behind their propulsion in cruising by developing scaling laws of their flight kinematics. Notably Strouhal number (St; ratio of flapping frequency times stroke amplitude to cruising speed) has been found to fall in a narrow range for animal cruising flights. However, St exhibits strong correlation to flight conditions; as such, its universality has been confined to preferred flight conditions. Since the leading-edge vortices (LEV) on flapping appendages generate the majority of propulsive forces, here we take the perspective of LEV circulation maximization, which generalizes the dimensionless vortex formation time to flapping flight. The generalized vortex formation time scales the duration of vorticity injection with the rate of total vorticity growth inside the LEV and the maximum vorticity allowed inside it. By comparing the new scaling with St of previously reported animal cruising flights of 28 species, we show that the generalized vortex formation time is consistent across different animals and cruising locomotion, independent of flight conditions. This finding advances the fundamental principles underlying the complex wing kinematics of biological flyers and highlights a unifying framework for understanding biolocomotion.

Across taxa, size, and biome, animals exploit their fluid environment to generate the propulsive forces required for flapping, swimming, and gliding. In many of these locomotory strategies, unsteady aerodynamic forces augment the animal’s propulsive force and efficiency (1, 2) via the roll-up of shear layers into swirling vortical structures. Indeed, recent studies have suggested that leading-edge vortices [referred to as leading-edge vortex (LEV) hereafter, Fig. 1; Movie S1] can account for up to 99% of the propulsive force in flapping flight (3), and vortex ring formation can enhance the flow impulse of jet-propelling systems by 50% (4, 5).

Fig. 1.

Path line visualization and schematic control volume of the development of the leading-edge vortex (LEV) around a flat plate airfoil at $Re\sim 750$. (A–D) Chronological path line visualization of the top surface of the airfoil. The LEV develops from a to c while attached to the airfoil. (A) Early phase of the LEV development. The airfoil has a chord length $c$. (B) Mid-phase of the LEV growth. Thickness of the LEV core, $\delta$. Inset, path lines near the trailing edge of the airfoil satisfying Kutta condition. (C) Fully grown LEV. Note the edge of the LEV is near the TE. (D) Postshedding stage. New trailing-edge vortex forms and lifts the LEV away from the airfoil. The fully grown LEV is shed by the trailing-edge after $\widehat{T}$ approaches 1. (E) Schematic control volumes ($C{V}_1$ and $C{V}_2$) around the airfoil at angle of attack $\alpha$. Dashed line, $C{V}_1$. Solid line, $C{V}_2$. Point A, stagnation point ($s_o$). Point B, leading edge ($s_{\mathit{LE}}$). Point C, trailing edge (TE). Arrow pointing from $C{V}_1$ to $C{V}_2$, diffusive flux. Arrow pointing from $C{V}_2$ to $C{V}_1$, convective flux. (F) Visualization of the bottom surface of the airfoil. Inset, stagnation point (yellow dot). All panels have the same airfoil with $c=25.4$ mm and $\alpha ={30}^{°}$.

Among the most extensively studied of these locomotory modalities are cruising flight and swimming, characterized by a periodic motion of propulsive appendages at a set frequency. Studies have made significant progress in deriving a universal rule governing the cruising propulsion across different spatial scales, flapping frequencies, and fluid media. Periodic propulsive motion is most commonly described using the dimensionless Strouhal number (1, 2, 6) ($St=fA/U$, sometimes referred to as the inverse of advance ratio) or reduced frequency ($f_r=fc/U$), where $f$ is the flapping frequency, $A$ is the motion amplitude of the wing or fin, $c$ is the chord length, and $U$ is the average body velocity. Both the Strouhal number and the reduced frequency are useful in quantifying unsteadiness in systems and represent nondimensionalizations of oscillatory phenomena; as such, both roughly measure the ratio of unsteady inertial forces to steady inertial forces. Studies have shown that the cruising Strouhal number often falls within a narrow range of 0.2 to 0.4 for a wide range of taxa, with length scales ranging from 0.1 m to 10 m (1). The narrow range of observed Strouhal numbers has been interpreted as the result of a fluid mechanical limitation providing strong evolutionary constraints on animal flight patterns.

However, Tobalske et al. (7) measures $St\approx 2$ for a slow-flying zebra finch, an order of magnitude greater than the frequently observed range of $0.2<St<0.4$. Further, the Strouhal number of the zebra finch appears to linearly depend on the flight speed Reynolds number (7) ($Re=Uc/\nu$, where $U$ is the flight speed, $c$ is the chord length, and $\nu$ is the kinematic viscosity of air). A strong dependence of Strouhal number on the cruising flight speed has also been observed in other flying animals, such as bats (8, 9) and bumblebees (10). While these deviations from $St\approx 0.3$ may be explained by nonethological experimental conditions, the dependence of Strouhal number on the flight conditions nonetheless challenges the notion of universality of Strouhal number.

One potential reason for the Strouhal number’s dependence on flight speed is that the velocity and length scales used to nondimensionalize the wingbeat frequency are weakly associated with the formation of the LEV. We hypothesize that a dimensionless number characterizing LEV formation would reduce the linear dependence of the dimensionless frequency (or timescale) on the Reynolds number.

In this regard, another dimensionless number, known as the vortex formation time (referred to as the formation time thereafter), $\widehat{T}$, has been used to describe the formation process of vortices (11). Generally, vortex formation time is defined as a time, $t$, nondimensionalized with a representative length, $L$, and a velocity, $U$, such that $\widehat{T}\sim Ut/L$. In the context of biological propulsion, the vortex formation process was first studied in jet propulsion systems (4, 12). It has been shown that pulsed jetting, which produces discrete vortices, can enhance jet propulsion by maximizing the circulation of each vortex (4, 12). This circulation maximization was shown to occur at around $\widehat{T}\sim 3$ to 5 (11, 13). These results highlighted that a vortex has an inherent limit on how much circulation it can accept.

The vortex formation time and the observed circulation maximization (i.e., growth limitation) have been extended to LEV formation (14–19). The leading-edge formation time, also referred to as advective time (17, 20), has also been shown to have a limit on its circulation growth. Although exact definitions vary among studies (14, 17, 19, 20), LEV formation time is often defined as a time nondimensionalized with a representative velocity (e.g., shear layer velocity) and chord length.

Because the LEV formation time uses velocity and length scales representative of the formation process, it is a promising dimensionless number for representing the unifying principle of flapping propulsion. However, its universality among flapping flyers has not been fully explored. One of the limitations in implementing the LEV formation time is the difficulty in incorporating the angle of attack. Organisms’ flapping vary not only in their morphology, stroke amplitude, and frequency but also in the angle of attack. Currently, there lacks a standard way of incorporating the angle of attack into the formation time definition. Some studies incorporate the angle of attack into the velocity scale (15, 17, 19) and some incorporate into the length scale (14, 21). Consequently, the incomplete incorporation of the angle of attack in the definition of LEV formation time prevents fair comparisons of the formation time among various flapping flyers.

In this study, we first generalize the definition of formation time that is central to the vortex limiting process. From this generalization, velocity and length scales incorporating the angle of attack are derived. Finally, we demonstrate that the resulting dimensionless number is consistent across cruising locomotion and only weakly dependent on the flight speed Reynolds number as compared to the Strouhal number. The introduction of a dimensionless number that is nearly invariant across diverse flyers operating in different flow regimes strongly bolsters the claim of universality of the physics underlying flapping flight.

Results

Generalization of Formation Time.

For flapping systems, formation time $\widehat{T}$ can be interpreted as a nondimensionalization of the stroke duration, $t$, wherein time is scaled by the representative velocity and length scales, $U$ and $L$, respectively:

$$\begin{array}{c}\hfill \widehat{T}=\frac{\mathit{Ut}}{L}.\end{array}$$ [1]

In a flow system with a single velocity and length scale, defining $\widehat{T}$ is unambiguous. For example, in the well-studied piston-cylinder vortex ring formation process (Fig. 2A), the only choices of $U$ and $L$ are the piston velocity and diameter of the jet exit, respectively. However, there are multiple velocity and length scales inherent to flapping systems, with various potential physical interpretations of $U$ and $L$ provided in the literature (4, 5, 11, 13, 22). The correct physical interpretation of $U$ and $L$ can aid in identifying physically relevant velocity and length scales in a complex case where such a selection is nontrivial.

Fig. 2.

Schematics of physical systems associated with vortex formation and of the flight kinematics. (A) Velocity scales, length scales, and representative vortices for (i) a vortex ring formed at the jet exit of a piston-cylinder apparatus and (ii) the LEV formed on the top surface of an airfoil. The length scale for the piston-cylinder case is the piston diameter $D$ and the associated velocity scale is the piston speed $U_p$. The length scale for the LEV case is the chord length $c$ and the relative wind $U_{\text{rel}}$, projected onto the airfoil, composes the velocity scales, $u_x$ and $u_z$, parallel and orthogonal to the airfoil, respectively. (B) Schematic illustration of flight kinematics of a biological flyer at the start of downstroke. The flyer is cruising horizontally in the $({\widehat{\mathit{e}}}_x,{\widehat{\mathit{e}}}_z)$ frame of reference at $U_{\infty }$. The wing flaps in a stroke plane at an angle $\alpha$ and wingtip speed $U_{\text{tip}}$. The flight speed and the wingtip speed combine into the relative wind $U_{\text{rel}}$, which is oriented at ${\beta }_1$ relative to the chord in the $({\widehat{\mathit{e}}}_{x^{\prime }},{\widehat{\mathit{e}}}_{z^{\prime }})$ frame of reference. The chord is oriented at a pronation angle ${\beta }_2$ in the frame of reference of $({\widehat{\mathit{e}}}_x,{\widehat{\mathit{e}}}_z)$. Inset, front view with stroke amplitude and shoulder distance. Organism of choice, zebra finch.

To accommodate such cases, we first generalize $\widehat{T}$ with an emphasis on the fundamental vortex formation process. Instead of scaling time with velocity and length, we scale time with the rate at which circulation $\mathrm{\Gamma }$ (i.e., vortex strength) grows inside the vortex bubble of interest, $d\mathrm{\Gamma }/dt$, and maximum allowable circulation in the growing vortex bubble of interest, ${\mathrm{\Gamma }}_{\text{max}}$:

$$\begin{array}{c}\hfill \widehat{T}=\frac{\frac{d\mathrm{\Gamma }}{\mathit{dt}}t}{{\mathrm{\Gamma }}_{\text{max}}}.\end{array}$$ [2]

The circulation growth limit is imposed by different physical processes, including vortex pinch-off from the shear layer (11) for vortex ring formation and LEV lift-off (18, 23, 24) for LEV formation.

In general, $d\mathrm{\Gamma }/dt$ scales as $U^2$ and ${\mathrm{\Gamma }}_{\text{max}}$ scales as $\mathit{UL}$ (13). Thus, the new definition recovers the original formation time definition for systems with a single characteristic velocity and length scale, while providing flexibility for more ambiguous systems. The velocity scale $U$ could also be interpreted as the shear layer strength. Indeed, shear layer strength indicates the convective transport of vorticity into the control volume (Fig. 2A), whose vorticity growth is represented by $d\mathrm{\Gamma }/dt$. The length scale $L$ in this definition could be interpreted as the “vortex limiting length scale” as it is related to the maximum circulation allowed in the flow. This physical interpretation of $L$ is different from what has been previously suggested [prior work has interpreted $L$ as the length of vortex sheets (13)].

To elucidate, we draw an analogy to filling up a water bottle. If a faucet with a constant flow rate, $dV/dt$, is to fill up a bottle of known volume, $V$, one can scale the time it takes to fill up the bottle as

$$\begin{array}{c}\hfill \widehat{T}=\frac{\frac{\mathit{dV}}{\mathit{dt}}t}{V_{\text{max}}},\end{array}$$ [3]

where $dV/dt$ is the faucet volumetric flow rate, and $V_{\text{max}}$ is the volume of the bottle. The critical dimensionless bottle filling time would always be one, when the bottle is full.

Similarly, if $d\mathrm{\Gamma }/dt$ is constant, the dimensionless time at which a vortex reaches maximum circulation is one. In practice, expressions for both $d\mathrm{\Gamma }/dt$ and ${\mathrm{\Gamma }}_{\text{max}}$ are not exact and time-varying. Then, a scaling factor which depends on the system’s physical configuration (13) can arise, such that $\widehat{T}$ is some constant of $O(1)$ but not exactly one. For example, in the piston-cylinder system, $d\mathrm{\Gamma }/dt\sim U^2/2$ (13, 25) and ${\mathrm{\Gamma }}_{\text{max}}\sim 2UD$ (26).

While the basis of the generalization is simple, determining how $d\mathrm{\Gamma }/dt$ and ${\mathrm{\Gamma }}_{\text{max}}$ scale when there are multiple velocity and length scales can be challenging. In the next section, we consider the vorticity flux from the surface of the airfoil to determine the velocity scale associated with $d\mathrm{\Gamma }/dt$ during flapping flights. Moreover, we extend the classical result of the Kutta–Joukowski lift theorem (27) to find a suitable velocity and length scale associated with ${\mathrm{\Gamma }}_{\text{max}}$.

$d\mathrm{\Gamma }/dt$ of LEV.

To determine $d\mathrm{\Gamma }/dt$, we considered the vorticity generation at the surface of an airfoil. In a control volume around a growing vortex bubble (Fig. 1E), the LEV is fed by vorticity originating from the underside of the airfoil from the stagnation point (Fig. 1; Movie S2) to the separation point at the leading edge. Thus, we can express the growth rate of the LEV as

$$\begin{array}{c}\hfill \frac{d{\mathrm{\Gamma }}_{\text{LEV}}}{\mathit{dt}}={\int }_{\mathit{CS}}\omega \left(\mathit{u}·\mathit{n}\right)dS,\end{array}$$ [4]

where $\omega$ is the vorticity, $\mathit{u}$ is the flow velocity, and $\mathit{n}$ is the outward unit normal of the control volume (SI Appendix, section 2.1).

In general, vorticity has three sources: pressure gradient on the solid surface, acceleration of flow relative to the solid surface, and vorticity annihilation (28, 29). For example, in the piston-cylinder system, vorticity is generated at the onset of piston acceleration at the boundary of the piston wall due to the acceleration of the fluid relative to the wall. The viscous effect then diffuses the vorticity away from the wall. Subsequently, vorticity is convected out of the piston as the boundary layer separates at the sharp cylinder exit.

Unlike the piston-cylinder system, the major contributor of the LEV circulation for an airfoil is the strong pressure gradient (see SI Appendix, section 2.3 for in-depth discussion) from the stagnation point under the airfoil surface (29, 30) ($s_o$, point A in Fig. 1E; Movie S2) to the separation point near the leading edge ($s_{\mathit{LE}}$, point B in Fig. 1E; Movies S1 and S2), which can be expressed as

$$\begin{array}{c}\hfill {\int }_{\mathit{CS}}\omega \left(\mathit{u}·\mathit{n}\right)dS\sim \frac{1}{\rho }{\int }_{s_o}^{s_{\mathit{LE}}}\frac{\partial \rho }{\partial s}ds=\frac{p(s_{\mathit{LE}})-p(s_o)}{\rho },\end{array}$$ [5]

where $p(s_{\mathit{LE}})$ and $p(s_o)$ are the pressures at the near-leading edge separation point and the stagnation point, respectively, and $\rho$ is the fluid density. Using steady Bernoulli’s equation, $d\mathrm{\Gamma }/dt$ can be expressed as

$$\begin{array}{c}\hfill \frac{d\mathrm{\Gamma }}{\mathit{dt}}=\frac{u{(s_{\mathit{LE}})}^2}{2}.\end{array}$$ [6]

This result is consistent with the previously suggested scaling (18, 31, 32). For potential flow over a 2D elliptical airfoil (27) at an angle of attack $\alpha$, the flow speed at the leading edge, $u(s_{\mathit{LE}})$, scales with $U_{\infty }\text{sin}(\alpha )=u_z$, such that Eq. 6 can be written as

$$\begin{array}{c}\hfill \frac{d\mathrm{\Gamma }}{\mathit{dt}}\propto {\left(u_z\right)}^2.\end{array}$$ [7]

The potential flow solution is an idealized flow model and accurate at the moment when the airfoil is impulsively started (33). In general, $d\mathrm{\Gamma }/dt$ would have an additional time dependence, such that the LEV circulation growth rate will change. However, a theory developed previously suggests that $d\mathrm{\Gamma }/dt$ at the onset has the most significant contribution to the final circulation of the LEV (18). Thus, the potential flow based initial vorticity production scaling, $u_z^2$ should provide a proper velocity scale. Moreover, previous experimental studies (15, 16) show that $d\mathrm{\Gamma }/dt$ scales with $u_z^2$, further supporting the scaling in Eq. 7.

Γ_max of LEV.

To find the scaling for ${\mathrm{\Gamma }}_{\text{max}}$ of the LEV, we use the classical result of Kutta–Joukowski theory. The maximum circulation around the elliptical airfoil that satisfies the Kutta condition — flow leaving the airfoil smoothly at the bottom and top surface at the trailing edge (TE)—is given by

$$\begin{array}{c}\hfill {\mathrm{\Gamma }}_{\text{max}}=\pi U_{\infty }\left(c+b\right)\text{sin}(\alpha )=\pi u_z(c+b),\end{array}$$ [8]

where $c$ and $b$ are the chord length and the thickness of the airfoil, respectively.

During the growth of the LEV, the streamline wrapping around the LEV bubble extends from the shear layer at the leading edge to the rear stagnation (reattachment) point on the airfoil (34). As the LEV develops, its core and the rear stagnation point moves downstream toward the trailing edge.

The Kutta condition at the TE (point C in Fig. 1E) is satisfied (Fig. 1B, Inset; Movie S1) until the rear stagnation point grows to the TE and the vortex starts to detach (Fig. 1D; Movie S3) at the TE (18, 34). Streamlines wrap around the airfoil and the growing LEV (Fig. 1E), such that the circulation inside the LEV is nearly equivalent to the thick airfoil satisfying the Kutta condition at the trailing edge, where the thickness, $\delta$, is approximated by the LEV thickness, ${\delta }_{\mathrm{LEV}}$. Here, we assume that the thickest LEV location along the chord coincides with the chord location of the LEV core. The LEV core location along the chord is denoted as $x_{\mathrm{LEV}}$, and the LEV thickness at the same location is denoted as ${\delta }_{\mathrm{LEV}}$. In contrast to the low-angle of attack flow without separation at the LE, $\delta$ is changing with time due to the LEV growth (chord length, $c$, remains constant). Our experimental data show that the geometric growth of ${\delta }_{\mathrm{LEV}}/x_{\mathrm{LEV}}$ is proportional to the ratio between chord-normal and chordwise velocities: ${\delta }_{\mathrm{LEV}}/x_{\mathrm{LEV}}\propto u_z/u_x\propto \text{tan}(\alpha )$ (SI Appendix, section 3).

We assumed the LEV circulation reaches maximum when the LEV core is at the center of the chord, i.e. $\mathrm{\Gamma }(x_{\mathit{LEV}}/c=1/2)={\mathrm{\Gamma }}_{\max }$. Combining ${\delta }_{\mathrm{LEV}}/x_{\mathrm{LEV}}$, $x_{\mathrm{LEV}}/c$, and Eq. 8, ${\mathrm{\Gamma }}_{\max }$ of the LEV can be expressed as

$$\begin{array}{c}\hfill {\mathrm{\Gamma }}_{\text{max}}\propto u_{z}c\left(1+\frac{1}{2}\frac{u_z}{u_x}\right).\end{array}$$ [9]

An alternative approach for finding ${\mathrm{\Gamma }}_{\max }$ scaling is to consider the tangential velocity along the streamline wrapping the LEV bubble (35). This approach yields the same scaling as Eq. 9 (SI Appendix, section 2.2).

$\widehat{\mathbf{T}}$ of LEV.

Combining the scaling of $d\mathrm{\Gamma }/dt$ and ${\mathrm{\Gamma }}_{\text{max}}$ of the LEV (Eqs. 7 and 9), the formation time $\widehat{T}$ can be expressed as

$$\begin{array}{c}\hfill \widehat{T}\equiv \frac{\frac{d\mathrm{\Gamma }}{\mathit{dt}}t}{{\mathrm{\Gamma }}_{\text{max}}}\propto \frac{{(u_z)}^{2}t}{u_{z}c\left(1+0.5·u_z/u_x\right)}=\frac{u_{z}t}{c\left(1+0.5·u_z/u_x\right)}.\end{array}$$ [10]

Here, $u_z$ represents the velocity scale that produces vorticity, and $c(1+0.5·u_z/u_x)$ represents the length scale that limits the vortex growth.

$\widehat{\mathbf{T}}$ of Animal Flight.

Based on 13 studies (7–10, 36–44) that reported animal flight kinematics of 28 species, we calculated $\widehat{T}$ according to Eq. 10 (SI Appendix). Across birds, bats, and insects, we found that $\widehat{T}$ reliably falls between 1 and 3. The constant $\widehat{T}\sim 2$ signifies that across different taxa and wide range of flight speed, biological flyers’ wing kinematics may be tuned to achieve maximum LEV circulation. The maximized circulation would lead to augmented lift production per stroke. Thus, this suggests that biological flyers comply with flow physics for a distinct aerodynamic advantage.

To test whether the formation time is better-suited to describe the physics underlying flapping flight than the commonly used Strouhal number, we compared $\widehat{T}$ with $1/St$ and $1/\widehat{T}$ with $\mathit{St}$ (Fig. 3). The coefficients of variation ($CV=\mu /\sigma$, where $\mu$ is the mean and $\sigma$ is the SD) of the dimensionless quantities are shown in Table 1. Both $\mathit{St}$ and $1/St$ have a relatively wider spread than that of $1/\widehat{T}$ and $\widehat{T}$, respectively ($C{V}_{1/\widehat{T}}/C{V}_{\mathit{St}}=0.26$ and $C{V}_{\widehat{T}}/C{V}_{1/St}=0.54$). Our statistical analysis suggests that the variation of $1/\widehat{T}$ and $\widehat{T}$ are both significantly smaller than that of $\mathit{St}$ and $1/St$ ($P\ll \alpha =0.05$).

Fig. 3.

Formation time ($\widehat{T}$) and Strouhal number ($\mathit{St}$) of the 28 species of biological flyers collected from the meta-analysis. (A) $\mathit{St}$ (green circles) and $1/\widehat{T}$ (orange squares) for individual species. The values of $\mathit{St}$ span two orders of magnitude while those for $1/\widehat{T}$ consistently lie in a narrow range between 0 and 2. Organisms are categorized as bats, birds, hummingbirds, and insects, respectively as shaded. (B). $1/St$ (purple circles) and $\widehat{T}$ (pink squares) for individual species. The values of $1/St$ and $\widehat{T}$ exhibit similar spread in order of magnitudes as those of $\mathit{St}$ and $1/\widehat{T}$. Organisms are categorized in the manner mentioned Above. (C and D) Visualizations of the distribution of $\mathit{St}$ and $1/\widehat{T}$ (C) and $1/St$ and $\widehat{T}$ in (D). The same color code in (A and B) applies. The corresponding shades represent the probability density of the two datasets. All data exhibit a single tailed non-Gaussian distribution, while $\mathit{St}$ and $1/St$ exhibit much higher variation than $1/\widehat{T}$ ($P={10}^{-5}$) and $\widehat{T}$ ($P={10}^{-3}$). Error bars in (A and B), one SD. Data used from refs. 7–10 and 36–44.

Table 1.

Mean (μ), SD (σ), and coefficients of variation (CV) of the four dimensionless quantities St, $1/\widehat{T}$, $1/St$, and $\widehat{T}$, and the corresponding one-sided $P$-values for each pair ($\mathit{St}$-$1/\widehat{T}$ and $1/St$-$\widehat{T}$) from the bootstrapping statistics ($\alpha =0.05$)

Variable	μ	σ	CV	P-value
$\mathit{St}$	0.60	0.64	1.07	$2\times {10}^{-5}$
$1/\widehat{T}$	0.54	0.15	0.28	–
$1/St$	2.79	1.70	0.61	$1\times {10}^{-3}$
$\widehat{T}$	2.02	0.53	0.33	–

Given the dependence of $\mathit{St}$ on flight speed Reynolds number ($\mathit{Re}$) that has been previously reported, $\widehat{T}$, $1/\widehat{T}$, $\mathit{St}$, and $1/St$ are plotted against flight speed $\mathit{Re}$ for species with especially large variation in $\mathit{St}$, including zebra finch, bat, hummingbird, and bumblebee (Fig. 4). In all cases, whereas Strouhal number has a dependence on Reynolds number (e.g., the approximately linear dependence of $1/St$ on $\mathit{Re}$ shown in Fig. 4B, D, F, and H), formation time shows a relatively flat relationship. To quantify the dependency of $1/St$ and $\widehat{T}$ on $\mathit{Re}$, we performed an interaction regression analysis using Ordinary Least Squares (OLS) method. We evaluated the one-sided $t$ test statistics on the interaction terms (difference between the dependence of $\mathit{Re}$ on $1/St$ and on $\widehat{T}$) from the interaction regression model. The corresponding $P$-values for the four animals were zebra finch, $P=1.26\times {10}^{-9}$ ($t=-19.67$); bats, $P=3.86\times {10}^{-24}$ ($t=-14.35$); hummingbirds, $P=3.24\times {10}^{-4}$ ($t=-5.40$); bumblebees, $P=9.11\times {10}^{-5}$ ($t=-5.32$), all of which are significantly smaller than $\alpha =0.05$. This suggests that $\widehat{T}$ is significantly less dependent on $\mathit{Re}$ (i.e., flight speed) than $1/St$. Thus, the proposed formation time exhibits both an overall reduction in variation compared to Strouhal number and maintains a relatively constant value across different species and cruising flight speeds.

Fig. 4.

Formation time ($\widehat{T}$) and Strouhal number ($\mathit{St}$) for the four representative animals. $\mathit{St}$ and $1/\widehat{T}$ are shown in the Top panels, whereas $1/St$ and $\widehat{T}$ are shown in the Bottom panels. (A and B), Zebra finch. (C and D), Bats. (E and F), Hummingbird. (G and H), Bumblebee. For all four animals, $1/\widehat{T}$ and $\widehat{T}$ exhibit nearly flat lines, implying independency on flight conditions ($\mathit{Re}$). However, values of $\mathit{St}$ and $1/St$ show an inverse proportionality and a linear dependence on $\mathit{Re}$. Zebra finch, $P=1.26\times {10}^{-9}$ ($t=-19.67$); bats, $P=3.86\times {10}^{-24}$ ($t=-14.35$); hummingbirds, $P=3.24\times {10}^{-4}$ ($t=-5.40$); bumblebees, $P=9.11\times {10}^{-5}$ ($t=-5.32$). Data used from refs. 7–10 and 36–44.

Discussion

Cruising flights have long enthralled physicists and biologists alike, given the complex fluid dynamics underlying locomotion and the musculoskeletal and neuronal systems required for such behaviors. Many versions of dimensionless numbers describing the relative effect of unsteady and steady inertial forces have been suggested to describe the rules governing the flapping locomotion of animals. Among these numbers, $\mathit{St}$ is perhaps most commonly used as the naturally intuitive scaling for periodically actuated flows. In this paper, we suggest that generalized $\widehat{T}$, a nondimensionalization of the time it takes for vortices to reach maximum circulation, is an appropriate alternative scaling to $\mathit{St}$. Vortex formation is ubiquitous and central to the generation of unsteady forces which provide crucial enhancement to the aerodynamic lift for fliers. We find that across many scales of flapping flyers, $\widehat{T}$ falls under a narrow range, much like $\mathit{St}$, but $\widehat{T}$ has significantly less variability than $\mathit{St}$ among the fliers (Fig. 3 and Table 1).

The universality of $\widehat{T}$ across regimes provides insight into the physics underlying the common goal in flapping in animals. Flapping flight appears to be characterized by kinematics that maximize the circulation of the leading-edge vortex while the LEV remains attached to the airfoil.

In our model, circulation reaches its maximum when $x_{\mathrm{LEV}}/c=1/2$. This is based on the assumption that the shape of the LEV bubble and thus the streamline encapsulating LEV and the airfoil is symmetric. In a more realistic case, this geometry may exhibit dependence on the angle of attack. Determining the shape of the streamline can further improve the ${\mathrm{\Gamma }}_{\max }$ estimation. One such approach is to consider the time history of the volume flow rate from the shear layer at the leading edge (34, 35). Interestingly, the thickness of the shear layer has been suggested to depend on the curvature of the streamline wrapping around the leading edge (34), which in turn depends on the angle of attack. This suggests that relaxing the assumption would result in estimated $x_{\mathrm{LEV}}/c$ would have angle of attack dependence, i.e., $x_{\mathrm{LEV}}/c\sim k(\alpha )$, where $k$ is a function of $\alpha$.

The benefit of maximizing LEV circulation can be gleaned from the Kutta lift equation (45): $L_{\text{Kutta}}=\rho \mathrm{\Gamma }U$, where the aerodynamic force increases linearly with LEV circulation. However, it is important to note that the Kutta lift calculated with the circulation of LEV is the total force that acts on both the airfoil and the LEV. The aerodynamic force on the airfoil alone would depend on the location of the LEV relative to the airfoil in addition to the strength of the vortex. Then the chord length could be limiting the aerodynamic benefit not because of the LEV circulation limitation but through the motion of the LEV providing reduced suction pressure. Coincidentally, the suction pressure provided by the LEV would be fully felt at around $x_{\mathrm{LEV}}/c=1/2$. Thus, the circulation at which the lift is maximized would remain the same, although the physical interpretation of ${\mathrm{\Gamma }}_{\max }$ would be modified to the circulation at which the LEV provides maximum aerodynamic force.

Nevertheless, both interpretations point to the same conclusion: A constant $\widehat{T}\sim O(1)$ across varying conditions indicates that flyers are fully exploiting to boost the unsteady aerodynamic forces provided by LEV formation in the vicinity of the wing.

The reduced variability in $\widehat{T}$ compared to $\mathit{St}$ results from its lack of dependence on the flight speed (Fig. 4). Whereas $\mathit{St}$ strongly covaries with flight speed (represented as $\mathit{Re}$), $\widehat{T}$ shows much weaker dependence on flight speed for zebra finch, bats, hummingbirds, and bumblebees. These four sets with experimental data available for large ranges of flight speeds all show that $1/St$ is linearly correlated with flight speed, resulting in order of magnitude variations in $\mathit{St}$ when cruising speed spans orders of magnitude. The covariance of $\mathit{St}$ and flight speed indicates that $\mathit{St}$ does not fully capture dynamic adjustments made by flyers in response to changing flight conditions.

The consistency in $\widehat{T}$ across flight speeds and species indicates that there may be substantial evolutionary incentive. However, it is unlikely that all wing kinematics parameters (stroke angle, pronation angle, fight speed, and flapping frequency) are tuned for the purpose of reaching target $\widehat{T}$. Instead, some parameters may be adjusted to maintain $\widehat{T}$ when others are changed in response to other demands (e.g., hypothetically, flyers may adjust pronation and stroke angle for the purpose of maintaining adequate lift or balancing drag; the flapping duration and amplitude could then be varied to achieve the target $\widehat{T}$).

The observed spread in $\widehat{T}$ among individuals and across taxa has several potential sources, beyond inherent biological variations in morphology and kinematics. Our scaling analysis is based on the 2-dimensional LEV formation during a translation of the fixed angle of attack airfoil. In reality, flapping flight consists of both translational motion of the wing and rotation of the wing. The circulation of the LEV has been reported to be attached to the airfoil for longer duration in rotating motions (46–49) and also in 3-dimensional translational motions (50). In both cases, vorticity transport in the spanwise direction stabilizes the LEV and prevents the detachment of the vortex. Thus, the incorporation of the velocity scale of the spanwise flow might further modify and improve the formation time.

Moreover, the LEV circulation growth is a temporally varying process. Previous studies have suggested the circulation growth scales with $t^{1/3}$ for intermediate angles of attack (51) and with $t^{1/2}$ angle of attack of 90^° (52, 53). Other studies have considered varying angles of attack due to the active or passive pitching of the airfoil (15, 16). In the future studies, to incorporate the temporal variation, Eq. 1 can be modified to $\int (d\mathrm{\Gamma }/dt)dt/{\mathrm{\Gamma }}_{\max }$, much like the temporally varying jet diameter (12) as well as pitching airfoil (15–17).

Last, the analysis presented in this paper assumes that LEV circulation maximization as the driving evolutionary goal. However, other evolutionary pressures may exist that play equal or stronger role in selection. For example, mosquitos are known to acoustically communicate with their wingbeats; maximizing the acoustic radiation efficiency may be a stronger selection pressure for mosquitos than maximizing circulation. Indeed, all efforts to derive a universal rule for biological cruising flight are complicated by the trade-off in complying with other evolutionary pressures in addition to meeting the propulsive need.

The result shown in this paper extends the circulation maximization idea that was first studied in a jet-propelling system (11) to a distinctly different flow. The two ideas appear to be equivalent when the generalized formation time definition is used. Perhaps, this brings us closer to a unifying theory for biolocomotions utilizing vortical structures (13).

Materials and Methods

Statistical Analysis.

We performed statistical analyses on the overall population ($n=110$) of the data and on the four specific biological flyers (zebra finch, $n=7$; bats, $n=42$; hummingbirds, $n=6$; bumblebees, $n=8$).

For the statistical analysis on the overall population, we performed a one-sided nonparametric bootstrap test. The coefficient of variation $C{V}_i={\mu }_i/{\sigma }_i$, where $\mu$ is the mean, $\sigma$ is the SD, and $i=St$, $1/\widehat{T}$, $1/St$, and $\widehat{T}$, were bootstrapped with ${10}^5$ iterations. The null hypothesis was $C{V}_{\mathit{St}}=C{V}_{1/\widehat{T}}$ and $C{V}_{1/St}=C{V}_{\widehat{T}}$. The corresponding $P$-values are shown in Table 1 and reject.

For the statistical analysis on the four individual flyers, we compared the dependencies of $\mathit{Re}$ on $1/St$ and on $\widehat{T}$. We performed $t$ tests with 95% confidence level on the fitting coefficients ${\beta }_i$, where $i=1/St$ and $\widehat{T}$, from the OLS model. The null hypothesis was ${\beta }_{1/St}\ll {\beta }_{\widehat{T}}$, meaning that $P<\alpha$ implies that $\widehat{T}$ is significantly less dependent on $\mathit{Re}$ than $1/St$ is.

The confidence level for all statistical analyses in this paper was $\alpha =0.05$.

Supplementary Material

Appendix 01 (PDF)

Dataset S01 (CSV)

Movie S1.

Path-line flow visualization above the airfoil during the initial development of the leading-edge vortex.

Movie S2.

Path-line flow visualization underneath the airfoil during the initial development of the leading-edge vortex.

Movie S3.

Path-line flow visualization above the airfoil during the later stage of the leading-edge vortex.

Data, Materials, and Software Availability

All study data are included in the article and/or supporting information.