Flight of Seeds, Flying Fish, Squid, Mammals, Amphibians and Reptiles 89

Flight of seeds, ﬂying ﬁsh, squid, mammals, amphibians and reptiles

A. Azuma University of Tokyo, Tokyo, Japan.

Abstract

There are four ways for dispersing flying seeds: parachuting flight with fine filaments or pappi and gliding, rocking and spinning (or autorotational) flights with a wing or wings. A bundle of pappi applied to small seeds of flowering plants enables the seed to drift in a wind at a low rate of descent less than 0.5 m/s, whereas the wing utilized widely for flowering trees makes the seed flyable at the rate of descent less 0.5 m/s for gliding seeds and near 1.0 m/s for spinning seeds. These values are not irrelevant to their environmental condition and season of dispersion. Flying fish and squid make gliding flight with a pair of pectoral fins and membraned legs, respectively. The aspect ratio of their wing is so large (more than 5) that they are high performance flyers without beating their wings. The flying fish obtains its take-off speed by the violent fanning of the lower part of caudal fin immersed in the water and by exposing other parts of body in the air to reduce the drag. On the other hand, the flying squid attains its take-off speed by the water jet expelled from a nozzle of mantle where the water is stored. During the flight, in responding to the speed reduction due to the drag, the squid increases the trimmed angle of attack or the lift coefficient and keeps the flying height constant. Flying mammals, amphibians and reptiles have a membranous wing, the aspect ratio of which is less than 2, and make the gliding flight with a poor gliding ratio. However, the low aspect ratio wing enables them to perform highly maneuverable flights and to land either on the horizontal ground surface without over-running or on the vertical tree surface.

1 Flying seeds

1.1 Pappose seeds

Fundamentally, in a calm atmosphere, pappose seeds make vertical descending ﬂights by utilizing the drag force acting on the pappi as shown in Fig. 1a and b. The aerodynamic characteristics of a circular cylinder, which can simulate not only a pappus but also a ﬂagellum of small swimming organisms in low Reynolds number, are given by Lamb [1], Gray and Hancock [2], Holwill and Burge [3], Cox [4], Chwang and Wu [5], Lighthill [6] and Azuma [7]. By conducting falling tests

WIT Transactions on State of the Art in Science and Engineering, Vol 3, © 2006 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) doi:10.2495/1-84564-001-2/2b Flight of Seeds, Flying Fish, Squid, Mammals, Amphibians and Reptiles 89

Figure 1: Parachuting flight and the aerodynamic forces acting on a pappose seed. of pappose seeds in calm air, Minami and Azuma [8] evaluated the well-fitted equation describing the drag force. As shown in Fig. 1b, under the assumption of no lateral (or side slipping) translation, the longitudinal and normal components of inflow velocity caused by the vertical descent w are, respectively, given by w sin β and w cos β. Then the longitudinal and normal components of the aerodynamic force acting on a pappus, which is assumed to be a fine circular cylinder with length of lp and located at azimuth angle ψ and coning angle β are, respectively, given by ft = µlpw sin β Kt, (1) fn = µlpw cos β Kn, where, according to Chwang and Wu [5] and Jones et al. [9], the coefficients Kt and Kn are selected to be = π − Kt 2 / loge 2lp/dp 0.5 , (2) = π + Kn 2 / loge 2lp/dp 0.5 . The lift and drag components of a pappus are given by l = f cos ϕ, (3) d = f sin ϕ, = 2 + 2 where f ft fn . Then, the drag force acting on all pappi and directing vertically upward is given by Dp = [( ft sin β) + ( fn cos β)], (4) n where the summation must be extended over every pappus arranged with respective azimuth angle ψ and the coning angle β.

This parachuting force must be balanced with the weight of the whole seed and the drag force of the seed and stalk. The seed and stalk are also assumed to be an ellipsoidal body and a circular cylinder, respectively. Their drag force is directed upward and is given by

1 D = ρw2S C + µl wK , (5) s 2 c D,s s t = 1 π 2 where Sc is the cross-sectional area of the seed body Sc 4 d and CD,s is given by Hoerner [10] as follows: 1/2 CD,s = 0.44(d/l) + 4Cf (l/d) + 4Cf (d/l) , (6) where the Reynolds number is based on the length of the seed. Then the vertical force is given by

W = D = Dp + Ds. (7)

This equation can be solved for the rate of descent for a given configuration of the seed. A dashed line designated ③ in Fig. 2 shows a relation of w = 7.0 ×(W/A), which represents approximately the above relation. In Fig. 2, results obtained from other numerical analyses, which are based on different expres- sions of the drag coefficient for the circular cylinder [8], are also presented by designations ①, ② and ④ as a function of either wing loading W/S or disc loading W/A. For comparison, many statistical data obtained from the flight tests of various species of pappose seeds are also presented in Fig. 2. Since their rate of descent is very low, less than 0.4 m/s, a slight wind can scatter them widely from the top of stem, the length of which varies in proportion to the height of surrounding grasses in order to be well ventilated.

Figure 2: Statistical data and numerical analyses of ﬂying seeds observed in the rate of descent versus disc loading (redrawn from [8]).

WIT Transactions on State of the Art in Science and Engineering, Vol 3, © 2006 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Flight of Seeds, Flying Fish, Squid, Mammals, Amphibians and Reptiles 91

1.2 Winged seeds

Winged seeds are further classified into three groups, making such flights as gliding, straying or rocking accompanied with some fluctuations of the center of gravity location, and spinning or autorotational flights, each of which utilizes the lift as well as the drag acting on a flattened wing.

1.2.1 Steady gliding If the lift and drag coefﬁcients are under a well-known relation = + 2 π CD CD0 CL/ AR, (8) the maximum lift-to-drag ratio can be given by 1 (L/D) = (C /C ) = πAR/C (9) max L D max 2 D0 at = π CL,(L/D)max ARCD0 . (10)

Then, under the assumption of large L/D, in the steady gliding ﬂight, the maximum range Rmax,s and the ﬂight velocity V(L/D)max from a given height H can be obtained approximately as follows: = · Rmax,s (L/D)max H, (11) = π V(L/D)max 2(W/S)/ρ ARCD0 . (12)

Similarly, under the same approximation, the minimum rate of descent wmin and other related equations can be given by w = 4 2/ρ W/SC1/4/(3πAR)3/4 = (4/3) (2/ρ)(W/S)C /πAR, (13) min D0 L,wmin = π 1/4 = 1/4 Vwmin (2/ρ)(W/S)/ 3CD0 AR V(L/D)max /3 , (14) at √ = π = CL,wmin 3CD0 AR 3CL,(L/D)max . (15) A species of winged seeds in Java, Alsomitra macrocarpa, shown in Fig. 3, makes an extremely high-performance gliding among other species. This is accomplished by its large aspect ratio wing, AR = 3 ∼ 4. The lift-to-drag ratio or gliding ratio is about L/D = 3 ∼ 4 and the rate of descent is w = 0.3 ∼ 0.7 m/s [11]. The stability of flight is guaranteed by having the following configuration [11]. The seed itself is very thin, about 1 mm in thickness, and core of the seed is located nearly at the center of gravity, which is a slightly forward position of the wing center. The wing is also very thin (from a few micrometers to some 10 µm) and has a swept and tapered plan form, twisted (washout) angle, reflected trailing edge, and adequately arranged position of the center of gravity. The reflected trailing edge must be essential for keeping longitudinal stability of tailless gliders [7]. Furthermore, when the seed is laterally slipped, the restoring force and moment can be generated by bending up the leading tip of flexible wing. It is interesting to know that the center of gravity of the winged seeds is located at an optimal position at which the respective seed can fly at the lowest rate of descent or the highest duration of flight [11].

Figure 3: A gliding seed with large aspect ratio wing, Alsomitra macrocarpa.

This species of ﬂying seed is a parasitic plant. It inhabits forests of the southeastern tropical area. Just before the rainy season in this area, many seeds take-off from a husk hanging from the tree which tangles with any branch of the parent tree. The winds surrounding the husk are not strong but include an upcurrent. Thus, the seeds can remain airborne for a long duration and disperse over a wide range even in the forest.

1.2.2 Spiral gliding Actually, the gliding flight of a low aspect ratio wing does not take a straight path but rather a spiral path accompanied sometimes by a short-period oscillation of rolling. The oscillation is caused by a nonlinear behavior of the local separation of flow, characteristic to the low aspect ratio wings, specifically, a delta wing at high angles of attack. This oscillation is known to be the wing rock in airplane flight dynamics and is caused by the lateral oscillation of vortex breakdown [12–14].

1.2.3 Straying In winged seeds, except the above Alsomitra macrocarpa, the gliding is usually performed with low aspect ratio wing and thus, the maximum gliding ratio or the maximum lift-to-drag ratio is low. The flight path of these seeds is very steep such that the gliding angle is γ = sin−1 (w/V)is ◦ = more than 35 and, the trimmed angle of attack αtrim αCM = 0 becomes large. In the case of Cardiocrinum cordatum shown in Fig. 4a, the trimmed angle of attack αtrim is approximately 13◦. By shifting the center of gravity by ±10% of the mean aerodynamic chord, the trimmed angle of attack shifts about ±7◦. If the center of gravity of Cardiocrinum cordatum moves backward by more than 10% of the mean aerodynamic chord, the trimmed angle of attack cannot be found and thus the wing is always unsteady. This condition corresponds to the random straying or rocking motion shown in Fig. 4b. Since the Reynolds number based on the flight speed and mean aerodynamic chord is less than 103 and, thus, the equation governing the flow (Navier–Stokes equation) is nonlinear, the flow pattern varies case by case depending on different initial conditions. During the flight, the flight mode is suddenly switched from one state to another easily, accompanied with the motion of alternating the leading edge from one side to the other

Figure 4: A straying seed with small aspect ratio wing and its ﬂight path.

Figure 5: Spinning samaras (data from [15]).

side, by either a little disturbance in the atmosphere or the unsteady motion of its own ﬂight characteristics. Speciﬁcally, in the case of Cardiocrinum cordatum, the plan view of which is a triangle, the falling mode is sometimes occupied by random straying or undulatory motion. This undulatory motion is stressed in the case of seeds such that the wing loading is low and the center of gravity is located at the middle of the wing [8]. Statistical data of the rate of descent of gliding and straying seeds are also presented in Fig. 2 as a function of wing loading.

1.2.4 Spinning flight A spinning or autorotational flight is, as shown in Fig. 5a–f, considered to be the ultimate state of spiral gliding with an extremely small radius of turn. This state of flight was analyzed theoretically and experimentally by Minami and Azuma [8], Azuma and Yasuda [15], Yasuda and Azuma [16], Yasuda and Azuma [17] and Azuma [18]. The most obvious differences between the gliding seeds and the spinning seeds are: (i) the seed itself and, thus, the center of gravity of the whole fruit is located at one side of the wing in the spanwise direction and the leading-edge side in the chordwise direction except for tulip tree, ash tree and Fraxinus griffithii; (ii) the wing area S is small, but the disc area A or sweeping area of the wing in autorotation is large. Thus, immediately

Figure 6: Force distribution on Acer diabolicum [15].

after launching, the wing initiates autorotational motion and soon attains a steady spinning. This is explained as follows. The vertical force distribution, n = l cos φ + d sin φ, horizontal force distribution consisting of driving and resistant force distributions, t = l sin φ − d cos φ, and angle of attack distribution α of an exemplified seed, Acer diabolicum Blume, in steady autorotational flight is shown in Fig. 6. The vertical force n is mostly concentrated near the three-quarter radius because the velocity against the wing chord is proportional to the radius, and the induced velocity increases (and thus the angle of attack decreases) near the wing tip. The spanwise integration of this force trims with the weight of seed. Since the driving force l sin φ is widely distributed along the span, whereas the resistant force d cos φ is, like the vertical force, concentrated near the three-quarter radius, the horizontal force t is negative near the tip and positive inboard of about 60% radius. Just after the take-off, the driving torque accelerates the spinning motion and soon it balances with he resistant torque and the spin attains a steady state. The stability about the feathering axis is kept by the inverse camber near the wing root. This is typically recognized in the phoenix tree shown in Fig. 5e. The coning angle of the wing is determined by the torque balance about the flapping hinge between upward moment generated by the normal force and the downward moment generated by the centrifugal force. In the cases of tulip tree, ash tree and Fraxinus griffithii, there are two components of spin axis: vertical and spanwise components, the latter of which results from a symmetrical chordwise mass distribution as can be seen in Fig. 5c. The statistical data of sinking rate in the spinning seeds are shown in Fig. 2. It is reasonable for these spinning seeds to have relatively high rates of descent near 1 m/s because, for example, in Japan, they grow at the latitude of 35◦ ± 5◦ N and the longitude of 135◦ ± 5◦ E (the east side of the continent of Asia) where the prevailing westerly is predominant, and the strong winds caused by the typhoons are observed frequently during the dispersing season for seeds. Usually, they do not launch themselves until an extremely strong wind blows.

2 Flying ﬁsh and squid

Figure 7 shows the typical configurations of flying fish and squids in gliding flight. Here let us consider the optimal flight trajectory in the case where gliding is not steady, i.e. flight is performed with variable speed, but the height is kept the same to enable utilization of the same initial energy.

Figure 7: Flying ﬁsh and squid.

Figure 8: Comparison of ﬂight paths between steady and unsteady gliding.

2.1 Flight at a constant height

From Fig. 8, it seems reasonable to derive the initial kinetic energy of unsteady gliding, 1 mV 2, 2 0 1 2 by equating the total energy composed of the kinetic energy in steady gliding, 2 m V(L/D)max , and the potential energy of the altitude difference, mgH, and to assume that the ﬁnal speed is determined by the same gliding speed, V(L/D)max . Then, the initial speed of unsteady gliding can be given by = + 2 V0 2gH V(L/D)max . (16) In unsteady gliding, by combining with the equations of motion, which are given by making γ = 0 in eqns (22a,b,c) written later, Azuma [7] and Kawachi et al. [19] obtained the range ratio between the maximum range of unsteady gliding Rmax,u and that of steady gliding Rmax,s as follows: = Rmax,u/Rmax,s loge F/ loge G, (17) where F and G, respectively, become 1 F = 1 + πARC (H/bµ ) + πARC (H/bµ )2, (18) D0 b 2 D0 b = π G exp ARCD0 (H/bµb) , (19) and where µb is the density parameter based on the wingspan,

µb = m/ρSb. (20)

Since, if the ground effect is out of consideration, the difference between F and G and thus between Rmax,u and Rmax,s is negligible in the reasonable range of H/bµb, the flying fish or squid can actually take either one of the above two courses: steady gliding or unsteady horizontal flight. However, if the value of H/bµb is large, such as in the case of an artificial (or high-performance) glider, the ratio Rmax,u/Rmax,s becomes small (<1). This fact shows that any artificial glider should use steady gliding flight to get the maximum flight range.

2.2 Unsteady ﬂight path

Let us define a performance function of the maximum flight distance between time t0 and time tf by tf J = V cos γ dt, (21) t0 which should be maximized for any initial condition at take-off. The equations of motion on the trajectory of flight are given by  ˙ =− + 1 2 + 2 π  V g sin γ 2 (ρS/W)V (CD0 µb CL/ ARe,H) , ˙ = − + 1 2 (22a,b,c) γ (g/V) cos γ 2 (ρS/W)V CL ,  Z˙ = V sin γ , where Z is the height from the ground and ARe,H is the effective aspect ratio obtained in the ground surface effect and is written as a function of nondimensional height H/b as follows [7, 19]:    AR out of ground effect (OGE),  = − 2 − 2 + ARe,H  AR/ (1/0.35 )(H/b 0.35) 1 (0 < H/b < 0.35),  (23a,b,c) AR (0.35 ≤ H/b),

The constraints for variables can be given by CL,min =−0.5 CL,min ≤ CL ≤ CL,max; (24a,b) CL,max = 1.4 Hmin = 0, out of ground effect (OGE), H ≥ Hmin; (25a,b) Hmin = b/10, in ground effect (IGE).

Initial and final conditions are given respectively by (V0, γ0, Hmin) and (Vf , γf , Hmin). The above variational problem can be solved for three unknown dependent variables (V, γ , H) and one unknown input CL. The results for a flying fish, Cypselurus heterurus doederlieini, are shown in Fig. 9 for the speed and the flight path, and in Fig. 10 for the input CL. These figures show that (i) as the take-off angle γ0 increases, the maximum range Rmax,u increases for out of ground effect (OGE) but decreases for in ground effect (IGE); (ii) near the final approach to the ditching, the lift coefficient CL is adjusted to the maximum value to allow jumping flight for OGE but to maintain the level flight for IGE; and (iii) the ground effect helps to increase the flight distance by about 20–30% over that in OGE flight. However, as shown in Fig. 11, for either OGE or IGE, the effect of the initial takeoff angle on the maximum range is so small that the maximum range can be obtained for any takeoff angle by making the flight height as small as possible after the take-off.

Figure 9: Flight path and speed variation (numbers show the speed V in m/s).

Figure 10: Change of lift coefﬁcient, U = 15 m/s [19].

In the flying fish the take-off speed can be attained by the violent fanning of the lower part of caudal fin immersed in the water and by exposing other parts of the body in the air to reduce the drag. In the flying squid, on the other hand, the maximum take-off speed in air is attained by the water jet expelled from a nozzle of the mantle where the water is stored inside.

Figure 11: Maximum range versus initial take-off angle [19].

3 Mammals, amphibians and reptiles

The membranous wing discussed in this section is usually of low aspect ratio and is, therefore, relatively rigid. Thus, the aerodynamic characteristics are quite different from those of the pterosaur and bats, which have high aspect ratio wings. As shown in Fig. 2 in [20], the low aspect ratio wing has the low lift slope but the highest maximum lift coefﬁcient, CL,max. The nonlinear lift, called vortex lift, of the low aspect ratio wings results from strong rolled-up trailing vortices as shown in Fig. 12.

3.1 Flying squirrel

Petaurista leucogenys shown in Fig. 13a is a squirrel belonging to Sciuridae. It lives in trees and executes gliding flight by extending a pair of patagia arranged between the fore and the hind legs on opposite sides of the body. The animal uses flight as a way of quick, economical and safe movement. It maneuvers in flight by changing the positions of its four legs and tail. The initial acceleration is obtained by kicking the tree with the hind legs. When landing on a tree, as shown in Fig. 13b, it flares its body to assume a large angle of attack and reduce its speed, so as to land on the vertical surface of the tree. The low aspect ratio of the wing enables the squirrel to maintain a relatively big aerodynamic force at slow speed at high angles of attack. The maximum lift-to-drag ∼ ∼ ratio can probably be given by (L/D)max = 3 at a flight speed of U = 10 m/s.

3.2 Other ﬂying animals

Several other kinds of flying mammals, such as the phalanger and the colugo are known to be good flyers. They all have membranous wings of furry skin extending between front and rear limbs. Specifically, during gliding flight, a flying lemur Cynocephalus, shown in Fig. 14a and b, beats its tail up and down in a fanning motion described in [20]. This locomotion mode is effective in reducing drag or improving gliding performance. Special gliding wings, in the form of membranes, have also evolved in certain frogs and geckos, and in the flying dragon, a kind of lizard [21, 22]. It is not clear whether the wings of these

Figure 12: Rolled-up trailing vortices.

Figure 13: Flight of ﬂying squirrel.

Figure 14: Flying lemur (photographed by RELA KOBO Co. Ltd.).

WIT Transactions on State of the Art in Science and Engineering, Vol 3, © 2006 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 100 Flow Phenomena in Nature animals are still in the developmental stage for further airborne life, or whether they are already well adapted for their present environment. A flying lizard shown in Fig. 15 for example, has a slender body with a dorsoventral flattening that is extended laterally as a slender wing. This slender configuration generates a large aerodynamic force based on the linear potential lift and nonlinear vortex lift; thus, it guarantees a low sinking speed for safe landing in spite of a steep descent path.

3.3 Flying snake

Flying snakes, belonging to the genus Chrysopelea, are tree-dwellers and are found in southern India, Burma, Malaya and East Indies. These snakes do not make clearly powered flight, but they can glide some far distances. When they wish to move quickly from a branch of a tree they flatten their circular–cylindrical bodies by swinging their ribs outward. Then, they glide to the ground or else to a branch or bush, whereupon the body again returns to the original circular–cylindrical shape [23]. However, as shown in Fig. 16, the ribbon-flat paradise tree snake, Chrysopelea paradisi, seems to have an amazingly thin and flattened body originally and zips through canopy airspace [24]. In flight, the potential lift is generated along the widening body from the head to the middle where its width is the maximum. A similar potential lift is generated at the top of the ski-board and the human body in jumping. The potential lift is further strengthened when the body takes a snaking form as shown in these figures or a part of the body is, as stated before, directed obliquely to the flight direction like the ski-jumping with V formation of ski-boards, so as to increase the width perpendicular to the flow or the added mass of the fluid and thus to increase the lift.

Figure 15: Flying lizard (courtesy of K. Kuribayashi).

Figure 16: Flying snake, ribbon-ﬂat paradise tree snake Chrysopelea paradisi (sketched from [23]).

The normal force caused by the drag due to the perpendicular flow component is also generated along the entire length of the body. This drag is further increased by the separated flow which is specifically observed at the sharp opposite sides of the body. Then, this kind of aerodynamic force is considered to be a kind of nonlinear aerodynamic force which is known as vortex lift for very slender body. According to Socha [25], the take-off began with the maximum upward acceleration of 14.4 m/s2 and the horizontal velocity of 1.7 m/s and, then, during the mid-glide, the snake laterally undulated at a frequency of 1.3 Hz, with a wave height of 33% snout–vent length. The air speed and sinking rate were 8.1 m/s and 4.7 m/s respectively, and the glide angle was 31◦ in the late part of trajectory. Its best glide ratio was 3.7, amazingly high. The snake is surprisingly adept at aerial maneuvering. It turns without banking. The turns are initiated by movement of the anterior body and occur only during the half of the undulatory cycle when the head is moving towards the direction of the turn [25]. The lateral undulation can help to increase the lift because the undulation adds some increment to the relative speed perpendicular to the body, the maximum speed increment of which is 0.86 m/s.

4 Conclusion

Various ways of gliding and spinning flights of living creatures are presented. The four ways of dispersing flying seeds are parachuting with fine filaments, gliding, rocking and spinning with a wing or wings. Their sinking rate is closely related to the environmental condition at the bearing season. Flying fish and squids jump out of water by a strong fanning motion of the caudal fin and a water jet from a nozzle of the mantle, respectively, and make almost horizontal flight at some specified height, probably, in order to avoid attacks by predators, fishes in the water and birds in the air. The flight range increases as the initial take-off speed increases and the flying height decreases. However, the correlation between the initial take-off angle and the maximum range is weak. A high gliding ratio is not necessary for flying squirrels and lemurs with low aspect ratio wings, but it is convenient to have good maneuverability for their ecology in the forests. In other animals having poor flying performance such as flying frogs, lizards and snakes, their flight seems to be useful to extend their activity in their habitats.

Nomenclature

= π 2 1 π 2 A disc area, A R or 4 b AR, ARe,H aspect ratio and effective aspect ratio obtained in ground effect as given in eqns (23a,b,c) b wingspan = 1 2 = 1 2 CL, CD lift and drag coefficients, CL L/ 2 ρV S, CD D/ 2 ρV S CD0 , CD,s minimum drag coefficient, drag coefficient of seed body Dp, Ds drag of all pappi, and stalk and seed d elemental drag of pappus, diameter of seed dp diameter of pappus G, F functions given by eqns (18) and (19) = 2 + 2 f elemental drag force, f ft fn fn, ft normal and tangential forces acting on pappus element g gravity acceleration

H height Kn, Kt drag coefficients of normal and tangential forces acting on pappus element in low Re number flow L lift L/D lift-to-drag ratio l elemental lift lp length of pappus m mass of body n vertical force of a blade element at a spanwise station r R flight range, radius of spinning seed r radial distance of a blade element S wing area = 1 π 2 Sc cross-sectional area, Sc 4 dp t time, horizontal force of a blade at a spanwise station r V flight speed W weight, W = mg w descending speed or sinking rate α angle of attack β coning angle γ flight path angle or gliding angle, γ = sin−1 (w/V) µb density parameter, µb = m/ρSb ρ air density ϕ angle defined by ϕ = tan−1 (d/l) φ inflow angle, φ = tan−1 (v/r) ψ azimuth angle spinning rate

Subscripts

0 initial state f ﬁnal state max maximum value min minimum value s steady state trim trimmed state u unsteady state

References

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