By Jenny Morber
People have long looked to nature to understand the secrets of flight. Leonardo da Vinci and the Wright brothers filled pages with observations on birds, Airbus now fits some jetliners with patches that mimic shark skin, and research groups are studying arrays of mini drones inspired by flies and bees. Yet only recently have we turned our attention to the lowly dandelion.
In 2018, researchers from the University of Edinburgh reported an interesting experimental finding on dandelion seeds [1]. They discovered that the soft filaments that radiate out from the dandelion seed stalk create a ring-shaped wake, which helps keep seeds aloft. Pier Giuseppe Ledda, Lorenzo Siconolfi, Francesco Viola, Simone Camarri, and François Gallaire extended that work by modeling the airflow around this structure, called the pappus [2]. Their study revealed the ideal number of filaments for maximum flight distance — the same number found in real seeds. It thus appears that the dandelion seed is optimized for steady cruising. This work represents a fascinating peek at an often-overlooked phenomenon and suggests the application of biologically-inspired designs for lightweight parachutes and aircraft instability challenges.
The team became interested in this mechanism when they observed an unusual separation between the top of the pappus and its air recirculation region (the area of circular flow behind impenetrable objects as they move through a fluid). Usually this vortex trails immediately behind. Blunt objects like trucks and barges have large recirculation regions, though designers strive for efficient aerodynamics to reduce the effect of the area of circular flow.
“When looking at the pictures of the dandelion seed that were produced by the group at Edinburgh, we were very surprised because while the seeds have a recirculation region, it is detached from the body,” Camarri said. “If you look just past the body, the flow is not yet reversed. It reverses at the given distance, which is very peculiar.” Camarri and his colleagues were able to attribute this anomaly to the porosity of the pappus. Like a parachute, the pappus must create enough drag to counterbalance the seed’s weight, but it gains stability by letting some air through. Exactly how this worked in a dandelion seed was anybody’s guess.
To tackle the problem, Ledda et al. combined linear stability analysis with an averaging technique for nonhomogeneous porous materials. They began by simplifying the pappus geometry. One can visualize the pappus as a wagon wheel and its filaments as thin cylindrical spokes. The area in the wheel’s center where the spokes meet is impermeable, but air can pass through the spokes in the outer region (see Figure 1). Varying the number of spokes or filaments \((n_f)\) and their diameter \((d_f)\) changes the porosity of the pappus.
Figure 1. Three depictions of a dandelion seed. 1a. Realistic sketch of a dandelion seed and pappus. 1b. A simplified discrete model of a spoke-like pappus, imagined as a wheel with spokes. 1c. An approximation in which the pappus is a porous disk with variable permeability. The continuous porous disk models flow, where porosity and permeability are functions of the disk radius. Figure courtesy of [2].
The researchers defined a cylindrical coordinate system \((x, r, \theta)\) with its origin at the disk center and \(r\) as the radial direction. The \(x\)-direction is parallel to the inflowing air velocity. The porosity of the pappus is then
\[\phi(r) = 1 - \frac{n_f d^2_f}{8tr},\]
where \(t\) is the disk thickness and \(r\) is the disk radius. The mean porosity is the ratio between the area of the disk’s spokes and the disk’s total area, written as
\[ \Phi = 1 - \frac{n_f d_f (D/2 - r_p) + \pi r^2_p} {\pi(D/2)^2}. \]
Here, \(r_p\) is the radius of the inner, impermeable region (the pulvinus) and \(D\) is the diameter of the entire disk. The researchers explored a wide range of mean porosities, keeping \(d_f\) constant while varying \(n_f\) (see Figure 2).
Figure 2. Plots depicting porosity distribution for different numbers of pappus filaments \((n_f)\) (2a) and permeability variation along the radius when \(n_f=100)\) (2b).\(\boldsymbol{Da}\) is a nondimensional permeability tensor. Figure courtesy of [2].
One can therefore treat the pappus as a nonhomogeneous rigid porous disk with an impermeable center. Ledda et al. performed a global stability analysis of the pappus using the unsteady incompressible Navier-Stokes equations:
\[\partial t u + u \cdot \triangledown u + \triangledown p - \frac{1}{Re} \triangledown^2 u = 0,\]
\[\triangledown \cdot u = 0,\]
where \(u\) is the velocity vector, \(p\) is the pressure field, and \(Re\) is the Reynolds number for fluid viscosity. The latter is defined as \(Re = U \infty D/v,\) where \(v\) is the fluid’s kinematic viscosity. In the porous region, average quantities describe the fluid’s motion according to a model based on a Brinkman formulation:
\[\frac{1}{\phi}\partial_t \boldsymbol{u} + \frac{1}{\phi^2}\boldsymbol{u} \cdot \triangledown u + \triangledown p - \frac{1}{\phi Re} \triangledown^2 \boldsymbol{u} + \frac{1}{Re} \boldsymbol{Da}^{-1}\boldsymbol{u} = 0, \]
\[ \triangledown \cdot \boldsymbol{ u} = 0 .\]
Here, \(\boldsymbol{Da}\) is a nondimensional permeability tensor. These equations are completed by imposing Dirichlet boundary conditions on the air inlet and lateral boundaries, stress-free requirements on the outflow boundary, and velocity and pressure continuity at fluid-solid interface boundaries.
Using a linear stability approach, the flow can be decomposed as \(q = Q_b(x, r) + \varepsilon q' (x, r, \theta, t)\), where \(Q_b = (U_b, p_b)\) is the “base flow” and \(q' = (u', p')\) is the unsteady perturbation with amplitude \(\varepsilon \ll 1\). This flow decomposition yields at zero order the base flow steady governing equations and \(\textrm{U}_{\textrm{br}} = \partial \textrm{U}_{\textrm{bx}}/\partial r = 0\) for axisymmetric solutions.
At order \(\varepsilon^1\) an unstable perturbation evolves, which expands in Fourier modes in the azimuthal direction \(q'(x, r, \theta, t) = q \hat{} (x, r) \exp (im \:\theta + \sigma t)\), where \(m\) is the azimuthal wave number and \(\sigma\) is a complex number with the perturbation growth rate as its real part and the frequency as its imaginary part. The resulting eigenvalue in the fluid region then becomes
\[\sigma \hat{\boldsymbol{u}} + \boldsymbol{U}_b \cdot \triangledown_m \hat{\boldsymbol{u}} + \hat{\boldsymbol{u}} \cdot \triangledown_0 \boldsymbol{U}_b + \triangledown_m \hat{\boldsymbol{p}} - \frac{1}{Re} \triangledown_m^2 \boldsymbol{\hat{u}} = 0,\]
\[ \triangledown_m \cdot \boldsymbol{\hat{u}} = 0,\]
and can be written as
\[ \frac{1}{\phi}\sigma \boldsymbol{\hat{u}} + \frac{1}{\phi^2}(\boldsymbol{U}_b \cdot \triangledown_m \boldsymbol{\hat{u}} + \boldsymbol{\hat{u}} \cdot \triangledown_0 \boldsymbol{U}_b) + \triangledown_m \hat{p} - \frac{1}{\phi Re} \triangledown_m^2 \boldsymbol{\hat{u}} + \frac{1}{Re} \boldsymbol{Da}^{-1}\boldsymbol{\hat{u}} = 0,\]
\[\triangledown_m \cdot \boldsymbol{\hat{u}} = 0\]
in the porous region. The operators are defined as
\[\triangledown_m p = \left[ \begin{array} \\\frac{\partial p}{\partial x} \\ \frac{\partial p}{\partial r}\\ \frac{im \: p}{r} \end{array} \right], \]
\[ \triangledown_m \boldsymbol{u} = \left[ \begin{array} \\ \frac{\partial u_x}{\partial x} \:\:\:\: \frac{\partial u_x}{\partial r} \: \: \: \: \: \:\: \: \frac{im}{r} u_x \\ \frac{\partial u_r}{\partial x}\:\:\:\: \frac{\partial u_r}{\partial r} \:\:\:\: \frac{im}{r} u_r - \frac{u_\theta}{r} \\ \frac{\partial u_\theta}{\partial x} \:\:\:\: \frac{\partial u_\theta}{\partial r} \:\:\:\: \frac{im}{r} u_\theta + \frac{u_r}{r} \end{array} \right] ,\]
\[ \triangledown_m \cdot \boldsymbol{u} = \frac{\partial u_x}{\partial x} + \frac{1}{r} \frac{\partial(ru_r)}{\partial r} + \frac{im}{r}u_\theta,\]
\[ \textrm{and} \] \[ \triangledown_m^2 \boldsymbol{u} = \triangledown_m \cdot (\triangledown_m \boldsymbol{u}) .\]
The researchers set the displacement mode to \(m=1\), which corresponds to the first instability in an axisymmetric wake past a circular flat disk set normal to the fluid flow [3]. This displacement mode requires axis regularity conditions
\[ \frac{\partial \hat{u}_r}{\partial r} = \hat{u}_x =\frac{\partial \hat{u}_\theta}{\partial r} = 0.\]
The open-source software FreeFem++ —with 200,000 degrees of freedom and boundaries at \(r \infty = 20,\) \(x - \infty = -25,\) and \(x +\infty = 50\)—yielded numerical solutions. The pulvinus is treated as a solid boundary with \(u=0\) on its border.
The team determined steady flow solutions \(Q_b\) for dandelion seeds with different numbers of filaments: \(n_f = 130\), \(100\), and \(50\), with \(Re\) fixed at \(400\). Given this analysis, solutions to the flow equations indicate vortex rings that are partially detached from the disk base. The recirculation region remains partially attached in the impermeable pulvinus region (see Figure 3).
Figure 3. Contour images showing steady and axisymmetric solutions to the flow equations for the following: 3a. \(n_f = 130 \: (\phi = 0.911)\). 3b. \(n_f = 100 \: (\phi = 0.931)\). 3c.\(n_f = 50\:\: (\phi = 0.964)\). The isocontours represent the velocity magnitude. Figure courtesy of [2].
\(Re\) is varied in the stability analysis. In the area of the disk, permeability increases as the number of filaments decreases, which affects the apparent fluid viscosity. Hence the local \(Re\) strongly depends on disk permeability. As \(Re\) increases, the wake shows two successive helical bifurcations: a steady initial bifurcation and a second one that is periodic in time.
Ledda et al. find a critical \(Re\), past which instability is constant and equal to an impermeable disk. This critical value increases to an average permeability \(\phi\) of approximately \(0.93\), at which the neutral stability curves diverge. Flow is thus even and linearly stable for porosity greater than \(0.93\), which occurs when the number of filaments equals \(100\). Interestingly, real dandelions typically carry \(100\) filaments; this suggests that a dandelion seed is optimized for steady and stable flight. Its filaments maximize drag until they begin to affect stability. “It’s very exciting to find in nature something that resembles an optimal principle,” Camarri said.
The researchers consider the pappus filaments to be rigid, but as anyone who has nuzzled a dandelion seed can attest, they are actually soft and pliable. Future studies could add more realistic refinements, such as accounting for filament deformation or modeling the seed in free flight with greater perturbations and lateral motion.
Ultimately, this work can help scientists study the dispersal of seeds or other similarly-sized objects in the air. Viscosity and length scales are important for applications based on these findings because small objects like dandelion seeds interact differently in the air than larger objects. The interplay between air viscosity and seed size makes this solution work. “It’s kind of magic,” Camarri said. “If you look at the volume occupied by the filaments with respect to the disk, we could more or less call it empty. Its ability to fly is therefore amazing.”
References
[1] Cummins, C., Seale, M., Macente, A., Certini, D., Mastropaolo, E.,Viola, I.M., & Nakayama, N. (2018). A separated vortex ring underlies the flight of the dandelion. Nature, 562(7727).
[2] Ledda, P.G., Siconolfi, L., Viola, F., Camarri, S., & Gallaire, F. (2019). Flow dynamics of a dandelion pappus: A linear stability approach. Phys. Rev. Fluids, 4, 071901.
[3] Meliga, P., Chomaz, J.-M., & Sipp, D. (2009). Unsteadiness in the wake of disks and spheres: Instability, receptivity and control using direct and adjoint global stability analyses. J. Fluids Struct., 25, 601.
Jenny Morber holds a B.S. and Ph.D. in materials science and engineering from the Georgia Institute of Technology. She is a professional freelance science writer and journalist based out of the Pacific Northwest.