Effect of wing fold angles on the terminal descent velocity of double-winged autorotating seeds, fruits, and other diaspores

Wind dispersal of seeds is an essential mechanism for plants to proliferate and to invade new territories. In this paper we present a methodology used in our recent work [Rabault, Fauli, and Carlson, Phys. Rev. Lett. 122, 024501 (2019)] that combines 3D printing, a minimal theoretical model, and experiments to determine how the curvature along the length of the wings of autorotating seeds, fruits, and other diaspores provides them with an optimal wind dispersion potential, i.e., minimal terminal descent velocity. Experiments are performed on 3D-printed double-winged synthetic fruits for a wide range of wing fold angles (obtained from normalized curvature along the wing length), base wing angles, and wing loadings to determine how these affect the flight. Our experimental and theoretical models find an optimal wing fold angle that minimizes the descent velocity, where the curved wings must be sufficiently long to have horizontal segments, but also sufficiently short to ensure that their tip segments are primarily aligned along the horizontal direction. The curved shape of the wings of double winged autorotating diaspores may be an important parameter that improves the fitness of these plants in an ecological strategy.

Figure 1

We parametrize the shape of the fruit and its sepals, here illustrated by a dried fruit and wings from the dipterocarp family where we extract the sepal length ($L$) and notice that the sepal has a curvature $K$ along the length of the wing here illustrated by fitting an osculating circle along the wing giving its radius of curvature $1/K$. The sepal's fold angle is defined as $KL$. The fluid has a density $\rho$ and a viscosity $\mu$. When the fruit is in free fall, it descends with a terminal velocity $U$ and autogyrates with an angular frequency $\mathrm{\Omega }=2\pi f$, where $f$ is the rotational frequency. Dimensional analysis gives us in addition to the fold angle $KL$, two nondimensional numbers that describe the flight; the Reynolds number (Re) is giving the ratio of inertia and the viscous force $\text{Re}=\frac{\rho UL}{\mu }$ and the Strouhal number (St) is giving the ratio of the rotational speed and the translational speed $\text{St}=\frac{fL}{U}$. Image courtesy of James Smith.

Figure 2

Our experimental methodology and work flow is composed of digitalization of the synthetic fruits by a 3D Computer Aided Design model (FreeCAD v0.16), which is then 3D printed and used for flight experiments in a water tank. After varying the weight $\mathit{mg}$ of each geometry by depositing a known weight of lead to its hollow fruit, we measure the terminal descent velocity $U$ and rotational frequency $f$. After data evaluation we return to the CAD design with guidelines for the wing design, i.e., most significantly in this study the fold angle $KL$.

Figure 3

The left part shows the raw data from our automatic tracking of the motion of a synthetic fruit, where we follow its wing tips and the vertical motion of the fruit. The lines are intended only as a guide to the eye to illustrate the motion of each individual wing. In the right part, we plot the vertical position $H$ of the fruit in one part of the tank and show that our synthetic fruits reach a steady terminal descent velocity $U$.

Figure 4

(a) Sketch of the fruit's wings that are decomposed into small elements $dl$ and moving with a frame of reference at the vertical descent velocity $U$. $\mathrm{\Omega }$ is the rotation rate, $dl$ the length of the blade element, $R$ the distance of the blade element to the axis of rotation, $\varphi$ the wing inclination at the location of the blade element, $U_N=Ucos\left(\varphi \right)$ the normal component of the descent velocity, $U_T=R\mathrm{\Omega }$ the tangential wind created by the rotation. (b) View of the blade element described in Fig. 4, in the plane perpendicular to the wingspan. $c$ is the chord of the blade element, ${\alpha }_p$ its pitch angle, $\theta$ the angle of the relative wind, $\alpha =\theta -{\alpha }_p$ the angle of attack, $U_R$ the wind speed “felt” by the blade element, $L$ and $D$ the lift and drag components, respectively, and $F_F$ and $F_N$ the resulting forward and vertical forces. The camber profile is illustrated by the curvature of the blade element.

Figure 5

(a) Parametrization used for the $C_L\left(\alpha \right)$ (solid line) and $C_D\left(\alpha \right)$ (dashed line) coefficients based on [43] using their Eqs. (12) and (13) but with $max\left(C_D\right)/2$. A much smoother stall behavior is modeled compared with high Reynolds number wings, in agreement with studies performed for rotating wings at moderate Reynolds numbers where a strong leading edge vortex is present. (b) The parametrization results in a smooth shape for the $C_L\left(\alpha \right)/C_D\left(\alpha \right)$ curve.

Figure 6

Eleven 3D-printed synthetic double winged fruits with $\psi =0^{\circ }, \mathrm{\Delta }\alpha =2.5^{\circ }$ and the same weight $mg=20$ mN, but different fold angle $KL$ are simultaneously released in the water tank. The image is a stroboscopic view of the path of the fruits and shows that the minimum terminal descent velocity is around $KL\approx 2.3$, in accordance with predictions of the blade element model [see Figs. 12 and 12].

Figure 7

We plot the experimental data for $\psi =0^{\circ },\mathrm{\Delta }\alpha =2.5^{\circ }$ for $KL=[1.7,2.3,3.0]$ where the wings span an area $A_d=[0.0058,0.0069,0.0061]m^2$, respectively. (a) The terminal descent velocity $U$ is plotted as a function of the wing loading $\sqrt{mg/A_d}$, where we see that Eq. (1) does not collapse the data onto a single curve with respect to the scaled curvature $KL$ where we want to note that the similarity of $KL=1.7$ and $KL=2.3$ is a special case. (b) The fruit's rotational frequency $f$ is plotted as a function of $\sqrt{mg/A_d}$, where $f$ is nonmonotonic with respect to the fold angle $KL$. (c) By rescaling $U$ and $f$ we plot the Strouhal number St, which is constant for each geometry but the magnitude of St is a function of $KL$.

Figure 8

Five 3D-printed synthetic double winged fruits with $\psi ={35}^{\circ }, \mathrm{\Delta }\alpha =2.5^{\circ }$ and with the same weight $\mathit{mg}=20\text{mN}$, but with different fold angle $KL$ are simultaneously released in the water tank. The image is a stroboscopic of the path of the fruits and shows that the minimum terminal descent velocity is around $KL=1.2$, in accordance with predictions of the blade element model [see Figs. 12 and 12].

Figure 9

We plot the experimental data for $\psi ={35}^{\circ },\mathrm{\Delta }\alpha =2.5^{\circ }$ for $KL=[0.06,0.6,1.7]$, where the wings span an area $A_d=[0.0053,0.0092,0.012]{\mathit{m}}^2$, respectively. (a) The terminal descent velocity $U$ is plotted as a function of the wing loading $\sqrt{\mathit{mg}/A_d}$, where we see that Eq. (1) provides a fairly good description of $U$ when the wings are not highly curved. (b) The fruit's rotational frequency $f$ is plotted as a function of $\sqrt{\mathit{mg}/A_d}$, where $f$ increases with the fold angle $KL$. (c) By rescaling $U$ and $f$ we plot the Strouhal number St, which is constant for each geometry but the magnitude of St is a function of $KL$.

Figure 10

2D contour maps for (a) the moment $M$, and (b) the resultant vertical force $F$ predicted by the blade element model Eq. (3) as a function of Re and St for the baseline fruit geometry, with a net mass loading of 1.2 g. The data presented in the figures are scaled of an arbitrary factor for ease of representation. The steady-state descent is obtained at $F=M=0$, and it is stable as discussed in the text.

Figure 11

(a) The cross section of the wing profile (left axis) corresponding to a synthetic fruit having fold angle $\mathit{KL}$ = 2.3 as a function of the distance $R$ between the blade element and the fruit's axis of rotation, and the corresponding angle of attack in the stationary regime (right axis). The part of the wing where significant lift is created is highlighted in gray, (b) the corresponding distribution of vertical force (left axis) and driving momentum (right axis).

Figure 12

(a) We solve the blade element model using the parametrization shown in Fig. 5, for the two series of seeds already presented in Figs. 6 and 8. The experimental data is shown by the markers, which illustrates that the fold-angle $KL$ is the essential parameter to obtain a minimal terminal descent velocity that would suggest optimal wind dispersion potential, as previously discussed. Good agreement is found regarding the general shape of the curves, the position of the optimal $\mathit{KL}$ for each base angle $\psi$, and the relative change of falling velocity observed between Figs. 6 and 8. (b) We show the Strouhal number St, for the same set of seeds. Here, satisfactory agreement is found when considering the simplicity of the model. Both the typical magnitude of St in each case, the effect of modifying the base angle $\psi$, and the general trend of St with respect to $\mathit{KL}$ are satisfactorily reproduced by the model.

Figure 13

(a) The terminal descent velocity and Strouhal number as a function of sepal camber ($\mathrm{\Delta }\alpha$) and sepal fold angle, as predicted by the blade element model for $\mathrm{\Psi }=0^{\circ }$. The addition of a camber does not affect the placement of the minimal descent velocity along the $KL$ axis, but only influences its magnitude. This is due to improved autogyration, as visible through the inset for the Strouhal number. (b) Changes in the base angle $\psi$ formed between the base of the wing and the vertical axis shift the placement of the minimal descent velocity on the $KL$ axis, which is moved to smaller $KL$ when $\psi$ increases. Simultaneoulsy, a larger $\psi$ leads to a larger projected wingspan which increases the relative wind created due to rotation on the outer part of the wings, which in turn reduces the equilibrium Strouhal number.

Figure 14

Parametrization for (a) $C_D\left(\alpha \right)$ and (b) $C_L\left(\alpha \right)$ inspired from translating wings at slightly higher Re numbers. A sharp stall behavior is present for an angle of attack of around $6^{\circ }$.

Figure 15

Comparison between experimental results with synthetic 3D-printed biomimetic fruits and the blade elements model, using the parametrization of Fig. 14. The results are in reasonable agreement for the terminal velocity [Reynolds number, panel (a)]. The results for the Strouhal number [panel (b)] follow the same trend as the experimental results, though the agreement is not as good as in Fig. 12, especially at higher values of $KL$ when stall takes place.

Figure 16

Influence of varying the peak value of $C_L\left(\alpha \right)/C_D\left(\alpha \right)$ for the parametrization inspired by translating wings. While the details of the curves shape are altered, the position of the optimum curvatures remains mostly unchanged.