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In this study, we use a thin rotating plate to generate propulsion and lift for a paper plate. And the thin plate rotates along the spanwise axis. We numerically determine the influence on aerodynamic characteristics with a rotational velocity of the thin plate. The rotational velocity is obtained with spin parameter which is the ratio of the peripheral speed of the plate to the main flow velocity. And the numerical simulations based on the discrete vortex method show that the autorotation mode of the plate in a uniform flow appears naturally when the spin parameter is unity. Vortex formed from the backward-rotating edge is weaker than those generated from the forward-rotating edge of thin plate. The maximum lift generated at S = 0.75 if S < 1. The negative moment becomes negative for the nondimensional rotating speed S ≤ 1.75. The most negative moment appears when S = 1; at that time, autorotation occurs naturally.

The purpose of this study is to understand the aerodynamic characteristics to generate propulsion and lift for a thin rotating paper plate. The aerodynamics of freely falling paper plate such as business cards was studied to understand the fluttering or the tumbling. We are focusing on the aerodynamics characteristics of autorotation of thin paper plate. The autorotation is free to rotate with a fixed axis of a centroid of paper plate. The displacement of paper plate during the autorotation is not only vertical but also horizontal. Moreover, the horizontal displacement is not a swing motion. This means that the paper plate has the direction of displacement with a horizontal. The paper plate during the autorotation generates the lift force [

From the previous study, the relation between shape of plate and the role of vortex for thin rotating plate was studied by Kubota et al. [

The main scope of our study is the generation of the lift force and the drag force due to the formation of vertex with the different rotational velocity of thin plate. The rotational velocity is obtained with the spin parameter which is the ratio of the peripheral speed of the plate to the main flow velocity.

In our previous study, we determined the aspect ratio of the plate by using a falling paper and observed that a plate with an aspect ratio of 7 achieved the longest flight distance, which meant it must have the highest lift/drag ratio. The numerical simulation allowed us to calculate the forces acting on the plate during rotation as a function of rotation speed. In addition, we discuss herein how to use a rotating plate to generate aerodynamic forces.

We performed numerical simulations for nonsteady-state forces acting on the rotating plate. The phenomenon in a falling rectangular paper strip which rotates naturally along its spanwise axis is known as autorotation [

Here N represents the revolution per minute of the rotating plate, c is the plate chord length. Also, the Reynolds number Re was used to determine the condition of flow. The rectangular plate with an aspect ratio 7 was used on the calculation, since the effective shape of plate was determined from our previous experimental investigation. Autorotation occurs at S = 1.0 according to wind-tunnel tests at Re = 9 × 10^{4} [^{3} to 1.23 × 10^{4} [

From the observation on our previous study, vortex formed by the autorotating plate was acting on the generation of lift force. We simulated the flow by using a vortex-blob method to observe events near the plate during rotation. The numerical simulation was implemented with the discrete vortex method. A similar simulation was performed by Iima, who simulated the flight of a butterfly by means of vortex blobs [

By numerical simulations, we also investigated the effects of the spin parameter S on the aerodynamic characteristics. The calculated forces are shown in _{x}, C_{y}, C_{m}, and R, respectively. In the range of S that we tested, the magnitude of the resultant force is a maximum at S = 0.1, and the extremum occurs from S = 1.0 to 1.25. The resultant force is not always large even if S is large. For example, the resultant force for S = 2 is smaller than that for S = 1. The force is a minimum at S = 0.6 if S < 1.

The drag shows tendencies which are similar to those of the resultant force. In contrast to the drag, the lift is a maximum at S = 2 and decreases to almost zero at S = 1.25 with decreasing S in the range from S = 2 to 1.25. The extremum occurs at S = 0.75 and becomes negative for S ≤ 0.25. The sign of the lift changes at S = 0.4, which indicates that the direction of the lift force is easily controlled by changing the rotation speed around S = 0.4.

It is interesting that the S of flapping insects is almost 0.5. For high maneuverability, we guess that insects evolved a ratio of flapping speed to flight speed that is near 0.4. The lift-generation mechanism for S > 1 is the same as the Magnus effect for a rotating cylinder. However, for S ≤ 1, lift is due to the difference of the pressures induced by vortices on both surfaces of the plate. The negative moment begins at the nondimensional rotation speed S ≤ 1.75. The maximum value of the negative moment occurs at S = 1; hence, autorotation occurs naturally. The moment of the force becomes negative except for S = 2. Thus, the plate rotates automatically, because the negative moment assists rotation. Namely, it is called the autorotation mode. It appears naturally at S = 1 because of the maximum absolute value of the negative moment.

The ratio of lift to drag for different values of S is shown in _{y}/C_{x}, the flight distance in the forward direction is maximum. For S = 1.25, which gives C_{y}/C_{x} = 0, the plate falls vertically. The flight path never deflects, because it does not generate lift in this case. Moreover, for S < 1, the maximum ratio C_{y}/C_{x} appears at S = 0.75, and this ratio becomes increasingly negative at S = 0.1 because of negative lift. A test plane with this rotating plate will fly backward.

On the other hand, for S = 0.5 - 1.0, the Lissajous loops are similar to each other. The size of the loop and the fluctuation become larger for S increasing within the given range. For S = 1.25 - 2.0, a protuberance appearing in the second quadrant grows with increasing S. The protuberance shows that negative drag (i.e., thrust) is generated at that phase. However, the drag is much larger than the thrust over a half rotation; therefore, we never notice the thrust generation at S > 1 when we observe the average force. The larger fluctuation results from the changes in the overall force that is produced at larger S. Thus, according to the simulations, we should adopt S = 1 for the effective rotating speed. We should make the plate easy to rotate so that an autorotation with S = 1 is naturally selected.

We investigate the forces acting on the rotating plate with the vortex method. The influence of speed on rotation of plate is considered by using spin parameter S. Vortices generated from the backward-rotating edge are weaker than those generated from the forward-rotating edge. However, the former move near the wing surface together with the plate; therefore, the induced surface pressure is larger. The lift attains a maximum at S = 0.75 if S < 1.

The moment becomes negative for the nondimensional rotating speed S ≤ 1.75. The most negative moment occurs at S = 1; thus, autorotation occurs naturally. The combination of lift and drag forces shows that the effective rotational speed for plate is S = 1. Since, the slower rotational speed (i.e., S = 0.05) causes the force generated unstably on the rotation cycle, and the rotating plate generates much larger drag than the thrust (or negative drag) during higher rotational speed (i.e., S ≥ 1.25).

S: spin parameter, πNc/(60U)

N: revolution per minute of the rotating plate; revolution/min.

c: chord length of rotating plate; mm

U: main flow velocity; m/s

C_{x}: drag force acting on the rotating plate; N

C_{y}: lift force acting on the rotating plate; N

C_{m}: moment of force; Nm

R: overall resulting force; N