Ash is common English name for Fraxinus genus plants tree, which is widespread across much of Europe, Asia and North America, often planted as shade tree. The seeds used in this paper were collected from the botanical garden of Institute of Botany, Chinese Academy of Sciences, and preserved in sealed bags to keep moisture. Eleven seeds that successfully enter the terminal stable autorotating and descending state in experiment trials were selected for further measurement and analysis. A typical sample of ash seed is shown in Figure 1.

As shown in Figure 1, ash seed exhibit a nearly symmetrical structure relative to the wingspan axis. Each sample seed’s mass was measured with an electronic balance with accuracy of 0.01 mg. The planer shape of these seeds was acquired using a scanner, thus the length, chord length and area of the winged seed can be obtained easily. Table 1 summarizes the morphological and kinematical parameters of all eleven ash seeds used.

A schematic diagram of the experimental apparatus is shown in Figure 2. Seeds were released ~2 m above the floor, which is high enough for the seeds to reach the stable terminal state. The observation section is surrounded by transparent and diffusive films to avoid any ambient disturbances. Also, diffusive films opposite

to the front and side cameras also make the backlit LED light sources (not shown in Figure 2 for the purpose of clarity) as even as possible. In order to clearly tell the rotating and spinning direction and correctly measure their angular velocities, three high-speed cameras were placed orthogonally and used to record the falling seed from the front, the side and the upwards directions, respectively. Each camera’s field of view (FOV) is about 40 × 20 cm, enough to cover several intact cycles of falling seeds based on their terminal descending speed as given in Table 1.

The governing equations of the flow around the flapping and rotary wings are the 3D incompressible unstable Navier-Stokes equations. The artificial compressibility method developed by Rogers [13] is used to solve the velocity and pressure. The governing equations in the curvilinear coordinate system are written as follows:

∂ Q ^ ∂ τ = − ∂ ∂ ξ ( E ^ − E ^ v ) − ∂ ∂ η ( F ^ − F ^ v ) − ∂ ∂ ζ ( G ^ − G ^ v ) + H G C L (1)

where Q ^ = 1 / J [ p             u               v             w ] T is the primitive variables, and J is the Jacobian determinant between the Cartesian coordinate system and the curvilinear coordinate system with the transformations ξ = ξ (x, y, z, t), η = η (x, y, z, t), ζ = ζ (x, y, z, t) and τ = t. The symbols E ^ = ( F ^     and     G ^ ) and E ^ v = ( F ^ v     and     G ^ v ) are the convective and viscous fluxes respectively. In the viscous fluxes, Re is defined as u R e = u ¯ c / ν where u ¯ is the reference velocity, which is defined as the mean velocity of seed tip, c is the chord length, and ν is the kinematic viscosity of fluid. For a moving/deforming mesh, the term H G C L is added to the right side of Equation (1) to enforce the geometric conservation law. A pseudo-time derivative of pressure is introduced into the continuity equation to solve Equation (1). This derivative uses the third-order flux-difference splitting technique for convective terms and the second-order central-difference scheme for viscous terms. The time derivatives in the momentum equation are computed using a three-point backward-difference implicit formula. Arithmetic accuracy is in second order for space and time.

Once the fluid field is solved numerically, integrating the pressure and viscous stress over the wing surface provides the total aerodynamic force acting on the wing. The vertical component of the total force is referred to as lift L, and the moment generated by the force component in the direction of the rotation is referred to as rotating moment Q. The dimensionless lift and rotating moment are referred to as lift C_L and rotating moment C_Q coefficients:

C L = L 0.5 ρ ( u ¯ ) 2 S (2)

C Q = Q 0.5 ρ ( u ¯ ) 2 S c (3)

where ρ is the air density and S is the wing area.

An O-H type mesh is used for numerical simulation (Figure 3). Before studying the aerodynamic forces and flow field of the wing model, CFD code [14], mesh density, the first mesh spacing, computation time step, and computational domain size used in this study have been validated, as shown in Figure 4, here, the time t ^ is non-dimensionalized by the period of spinning around wing span axis. As a result, a numerical solution independent of mesh and time steps can be achieved when the mesh dimension is 70 × 75 × 152 (in the normal, chordwise, and spanwise directions, respectively), the domain size is 30c, the first mesh spacing at the wall is 0.001c, and 400 time steps are used in one spinning cycle.

To describe wing kinematics, two coordinate systems are introduced here (see Figure 6), the earth frame (oxyz) and the wing-fixed frame (ox_wy_wz_w). The z_w axis is along wing span and the x_w axis is along the chord line pointing to leading edge. The kinematic parameters are the descending velocity (v_d), rotational speed about vertical axis (ω), spinning speed around wing span axis (ω_f) and

coning angle (θ). The coning angle (θ) is defined as the angle between the wing span axis (z_w) and the horizontal plane (xz), the pitch angle (α) is the angle between the chord line of the seed (x_wz_w) and the horizontal plane (xz). Table 1 gives the kinematic and morphological parameters.

The time history of the lift coefficient is shown in Figure 7(a). The periodical lift shows that the ash seed reaches a stable state and experiences two times variations of lift within one spin cycle. When the upward surface flips downward, the lift varies in a period. It can also be seen that the time course of lift is similar to a sinusoid curve and the peak C_L reaches about 1.5. The cycle-averaged lift coefficient (C_Lavg = 0.889) is about 10% larger than the seed non-dimensional weight ( G * = m g / 0.5 ρ ( u ¯ ) 2 S = 0.806 ), which indicates that the seed weight is balanced by the aerodynamic force pretty well, enabling the ash seed to descend at a relative low speed (see Table 1, v_d = 1.114 m/s).

Figure 7(b) presents the aerodynamic moment about the vertical axis. Although the maximum and minimum moment in one span spinning cycle are not symmetric about zero, the cycle-averaged moment is very close to zero, implying that rotating about vertical axis also reach a stable state. When the wing surface rotates from horizontal to vertical, the moment direction is opposite to the vertical rotation, which works as a drag preventing the rotation; while the wing surface flips from vertical to horizontal, the moment direction is same as the vertical rotation, indicating that the wing aerodynamic force plays a role of thrust driving the wing to rotate. Overall, the moment driving the wing to rotate around the vertical axis is generated when the wing surface flip from vertical to horizontal, while the damping moment is generated when the wing surface flip from horizontal to vertical.

Figure 8 gives the contour plot of vorticity at span location of 60% wing length from wing root. At time t₁, the wing generates maximum lift and the leading-edge vortex (LEV) is formed and remains attached to its upper surface, while the trailing edge vortex is also quite strong but separates from the wing surface. At time t₂, the wing pitches up and the LEV starts to separate from the wing surface, correlating with a dramatic decrease in lift. When the wing pitches

almost upright ( α = 84 ∘ ), the lift decreases to its minimum value. It is obvious that from t₁ to t₃ the horizontal component of aerodynamic force (F) is drag, meaning its direction is opposite to rotating motion. On the other hand, while from t₃ to t₅the horizontal component of aero dynamic force is thrust. As a result, the cycle-averaged moment C_Qavg is almost zero due to its periodical essence.

As aforementioned, due to rotations about vertical axis and span axis, the seed can generate enough aerodynamic force to balance its weight, which allowing a low speed descent. However, the lift is not through the center of mass, therefore there exist aerodynamic moments acting on the mass of center. To explain this, another frame ( o x ′ y ′ z ′ ) has to be introduced, whose z ′ axis is in the same direction as the z w axis and x ′ axis always in horizontal, thus the frame ( o x ′ y ′ z ′ ) only rotates about y axis as the wing rotates. As a result, the spin-cycle-averaged aerodynamic moment will be parallel to x ′ axis. And the spin-cycle-averaged total angular moment ( L ) will be in o y ′ z ′ plane (see Figure 9), which can be determined as:

L = I z ( ω 1 + ω f ) + I y ω 2 (4)

where I y and I z are moment of inertia about y ′ and z ′ , ω 1 and ω 2 are the components of ω in z ′ and y ′ , ω f is the Euler angle rate ( α ˙ ) about z ′ . Therefore, the z ′ component of angular moment ( L 1 = I z ( ω 1 + ω f ) ) is much smaller than that of y ′ component of angular moment ( L 2 = I y ω 2 ), because I y is about two order larger than I z while ω 1 , ω 2 and ω f in same order. Therefore, the total angular moment ( L ) is almost perpendicular to axis z ′ in o y ′ z ′ plane (see Figure 9). It should be noticed that the cycle-averaged moment due to aerodynamic forces is perpendicular to the total angular moment (the averaged moment about y axis nearly zero, see Figure 7(b)), thus only driving the total angular moment vector to process (rotating about vertical axis y ) but not change its magnitude. It can also be seen that the time rate of cycle-averaged total angular moment is:

d L / d t = ω × L (5)

which is pointing to the same direction of the aerodynamic moment. This processional movement about vertical axis is very similar to the processional movement of a spinning top on a table whose moment is caused by gravity, except that the total angular moment of ash seed is not along the spin axis but almost normal to it (Figure 9).

The contour plots of z w component of vorticity at different spanwise position from non-dimensional time t 1 to t 4 are given in Figure 10, as well as wing surface pressure distributions. It can be seen that, at time t 1 ,from wing root to tip, the LEV remains attached, therefore a lower pressure occurs on the upper surface near leading edge; at time t 2 , as the wing spins, the LEV and TEV are all detached, the lower pressure on the upper surface increases, correlating with a decrease in aerodynamic force; at time t 3 , the pressures of both surfaces become almost same, thus a lowest force is generated; at time t 4 , the pressure of upper surface are not as much lower as that at time t 1 but it covers a relative large region, and the pressure of lower surface also becomes more smoother, thereby generating almost same forces as time t 2 .