Fundamental Analysis of Three-Dimensional ‘Near Fling’

The ‘fling’ mechanism was discovered by Weis-Fogh (1973) through observing the flight of the wasp Encarsia formosa. The mechanism was paid increasing attention because a much larger force than the quasi-steady-state value was obtained by observation of the flight of insects that used these mechanisms. Lighthill (1973) analyzed this mechanism theoretically by using inviscid theory. He ignored the vortices shed at the outer edges. Maxworthy (1979) performed a series of experiments and pointed out that the magnitude of circulation generated during ‘fling’ is much larger than that calculated by Lighthill. Edwards and Cheng (1982) and Wu and Hu-chen (1984) introduced a concentrated vortex shed at the outer edge into an inviscid flow analysis and indicated the importance of this vortex. Haussling (1979) solved the two-dimensional Navier–Stokes equations at Re=30 and obtained the instantaneous streamlines and vorticity lines.

Spedding and Maxworthy (1986) measured the instantaneous lift forces on a pair of wings in ‘fling’. Ellington (1984a) pointed out that ‘near fling’ motion is also observed in the flight of some insects. In these studies, two-dimensional ‘fling’ and ‘near fling’ were assumed, but three-dimensional effects were not considered except by Maxworthy. Quantitative study of three-dimensional motion is important in analyzing the actual flight of an insect.

This paper describes both experimental and theoretical studies of three-dimensional ‘near fling’. The distance between rotation axes of the plates of a pair was varied in order to make clear the interference between the two plates. ‘Fling’ is regarded as a limit of ‘near fling’ where the distance between two rotation axes becomes zero. The hydrodynamic characteristic of ‘fling’ can be extrapolated from the experimental results of ‘near fling’. A comparison of the present three-dimensional analysis with the previous two-dimensional analysis for beating wings (Weis-Fogh, 1973; Ellington, 1984b) is presented. The results of the comparison will make clear the limitation of the applicability and the accuracy of the two-dimensional analysis that has been used to analyze beating wings. The large quasi-steady force coefficient obtained by two-dimensional analysis (Dudley, 1991) is verified by the present research. The numerical calculation technique developed in this paper makes it possible to analyze real insect flight where ‘fling’ or ‘near fling’ is used.

The experimental apparatus is shown in Fig. 1. An acrylic plate was suspended from a load cell in front of a mirror in a water tank (600mm×600mm×800mm). The mirror reflected an image of the plate. This made it possible for the flow around the plate in front of the mirror to simulate the flow around the two plates of a pair. A verification of this experimental approach, a comparison with an experiment using a real pair of rotating plates, is given in the Appendix. Two different shapes of plates, rectangular (plate A) and triangular (plate B), were used. A small weight was attached to each plate in order to create an initial moment. The configuration of the plate and the weight are shown in Fig. 2, and the dimensions of the plates and the weight are given in Table 1. The geometrical relationship between the mirror and the plate is shown in Fig. 3. The non-dimensional distance between the rotation axis of the plate and the mirror, d=d₀/x_tip, was varied in order to investigate the interference between the two wings of a pair in ‘near fling’ motion. In addition, the effect of varying the initial opening angle α₀ was studied. The test cases are shown in Table 2.

The plate was held in place by a stick before the initiation of motion. The stick was then removed and the plate began to rotate around the axis by gravitational force. The resultant variation of the opening angle α was recorded by a video camera. The angular velocity and angular acceleration α were obtained by differentiating α and ⁠, respectively. The forces in two directions, F_X and F_Z, and moment, M_Y=F_XL, were measured by the load cell as shown in Fig. 3. The value M_Y/L was compared with the directly measured force F_X to confirm the accuracy of the values of F_X.

Edwards and Cheng (1982) and Wu and Hu-chen (1984) pointed out that forces acting on plates in ‘near fling’ motion are strongly influenced by vortices shed from the outer edges of plates. In order to make clear the structure of the vortex system around a plate in ‘near fling’, two kinds of flow visualization tests, three-dimensional and two-dimensional, were performed. The three-dimensional test is more realistic in simulating the flow around an insect wing. The two-dimensional test is effective for understanding the fundamental structure of the vortex system. The two-dimensional test also provided results that could be compared with results of previous studies, which used two-dimensional models (Ellington, 1984a; Spedding and Maxworthy, 1986).

Plates of two different shapes, plate A and plate B, were used in the three-dimensional test. Two different distances between the plate and the mirror, d=0.02 and d=∞, were set for each plate shape. The experimental apparatus is shown in Fig. 1. The pictures of flow were taken by two still cameras, cameras 1 and 2. The wake was made visible by use of the dye Methylene Blue (Sunada et al. 1989).

Fig. 4A,B shows photographs for the cases of distance d=0.02 and d=∞, respectively, for plate A with the initial opening angle α₀=0. The photographs taken by camera 1 and by camera 2 show the flow pattern at α≈2 and α≈0.7, respectively. There is little difference of vortex systems between the two distances, d=0.02 and d=∞. A sketch of these vortex systems is shown in Fig. 5A. It can be seen that the vortices are generated from all four edges. The vortices from the edge indicated by the shaded area in Fig. 5A moved together with the plate and were always located near the plate surface. The vortices from the other edges were left in the water. It is estimated that the vortex near the rotation axis, the inner vortex, disappears. This is because it is cancelled by the inner vortex from the other (e.g. the image) plate when the distance d becomes zero. One vortex ring is, then, generated from the pair of rectangular plates.

Fig. 4C,D shows the results for the cases of distance d=0.02 and d=∞, respectively, for plate B. There is little difference in the vortex systems at the two distances, d=0.02 and d=∞. A sketch of these vortex systems is shown in Fig.5B. Again, it is observed that the vortices generated from the inner edge indicated by the shaded area in Fig. 5B move together with the plate. The vortices from the outer edge are left in the water. When the distance d becomes zero, two vortex rings are generated and they are linked to each other at the connecting point of the two triangular plates.

The flow pattern in the two-dimensional test was made visible by the use of aluminium dust floating on the water surface. The experimental apparatus is shown in Fig. 6A. A test plate, the chord length of which is 60mm, rotates around an axis which is perpendicular to the water surface. The non-dimensional distance between the rotation axis and the wall is d=d₀/c_v=0.4. Pictures of the flow pattern were taken by a still camera set above the water tank. The rotation speed Ω was about π/2rad s⁻¹ and the Reynolds number, Re=0.75c_v²Ω/𝒱, was 3×10³. The result is indicated in Fig. 6B. The vortices from an inner edge were always located near the surface of plate, but the vortices from an outer edge were left in the water. This behaviour of vortices was commonly observed in the results of the three-dimensional test (Fig. 5A,B). The same behaviour of vortices from the outer edge was also observed in the results of two-dimensional ‘fling’ (Spedding and Maxworthy, 1986). The behaviour of the vortices is different, however, from that in the sketch of two-dimensional ‘near fling’ by Ellington (Fig. 4c of Ellington, 1984a), where the inner vortices were left in the water and the outer vortices were located near the plate surface.

A model was developed to calculate the entire time histories of the normal force and moment acting on a pair of three-dimensional triangular plates rotating symmetrically around an axis. The normal force is drag because it is parallel to the inflow velocity due to the plate rotation.

Fig.8A–D shows the motion of plates that occurred when the initial opening angle α₀ was 0. Time histories of opening angle α for plates A and B are shown in Fig. 8A and B, respectively. Fig. 8C and D indicate, respectively, how and varied with α for two specific cases. The results for other cases are located between these lines. It is observed that the variation of the angular acceleration is large at the beginning of the motion and that the results of the angular velocity for each of the cases are similar to each other at about ⁠.

Fig. 9A–C shows the motion of plates with α₀=π/2 for two specific cases. It is observed that there is little difference among all the cases at t=0.

The measured and calculated forces and moments acting on a plate are indicated in Fig. 10A–D for α₀=0 and in Fig. 11A,B for α₀=π/2. The shaded areas in Fig. 10A and Fig. 11A show the normal force F_N and the tangential force F_S, which were calculated with equation 1 using the measured values of F_X and F_Z obtained from the load cell. The results of all the experimental cases are located within this shaded area. It is observed that the difference is small among the cases. The bold lines in Fig. 10B and Fig. 11B show the experimental results of the relationship between α and M_f, which is the same for both plate A and plate B, as stated before. The solid lines in Fig. 10A,B and Fig. 11A,B show the analytical results for test cases with d=0.02, and the broken lines in Fig. 10A,B show the analytical results with d=∞. The total values of the normal force and the moment, F_N and M_f, are compared with the components, F_N1 and M_f1, which are caused by the dynamic pressure. It is observed that the total values obtained by calculation are in good agreement with the experimental results for both cases. Therefore, it is verified that this calculation technique provides a means of estimating the normal force and the moment around the rotation axis, which act on a triangular plate during ‘near fling’ with various distances d.

The chain lines in Fig. 10A,B and Fig. 11A,B show the results obtained by the numerical calculation, where the measured plate movement for d=0.02 is assumed and the induced velocity generated by the other plate is ignored. A comparison of the chain lines with the solid lines reveals the interference effect between the two plates of a pair. It is observed that this interference effect is very small when the opening angle α is near π/2.

Fig.10C,D shows the circulation of the bound vortex which is generated along y=−c_tip/4 at α=0.3, and each component of the normal force indicated in equation 12. In these figures, the solid, dotted and broken lines indicate the components of the bound vortex circulation or the normal force. The width of a line indicates the calculation model. The heavy lines express the results obtained by using the complete vortex model with d=0.02. The narrow lines express the results obtained by using the calculation model, where the measured plate movement for d=0.02 is assumed and the induced velocity generated by the other plate is ignored. The difference in width between similar lines, solid, dotted or broken, indicates the effect of the interference between plates of a pair. It is observed that this interference increases the values of Γ_C near the rotation axis. Because the timewise change of integrated Γ_C along the x-axis is equal to the circulation of the vortices shed into the flow, the interference increases this circulation. The induced velocity component along the surface of the plate is made larger by the stronger circulation of both the vortices generated from the plate and Γ_C by the other (e.g. the image) plate. The increment of the induced velocity component is suppressed by the stronger circulation of the vortices generated from the other plate. As a result, the induced velocity component along the surface of the plate is made larger by the interference between plates of a pair. The bound vortex circulation Γ near the rotation axis on the plate is also increased by the interference. Hence, the value of F_N1(C,C)+F_N1(C,NC) is increased, because the value of F_N1(C,C)+F_N1(C,NC) is proportional to the multiple of the bound vortex circulation Γ and the induced velocity component along the plate surface. The other component of normal force, F_N1(NC,NC), the value of which is zero for an isolated plate as indicated in equation 13, has a positive value. Therefore, the interference increases the normal force due to dynamic pressure, F_N1(C,C)+F_N1(C,NC)+F_N1(NC,NC), and the moment around the rotation axis, M_f1(C,C)+M_f1(C,NC)+M_f1(NC,NC). This is indicated as the difference at α=0.3 between the solid line and the chain line in Fig. 10A,B. When the opening angle α increases, this interference between the two plates of a pair becomes small. It is also observed in Fig. 10D that the value of F_N1(C,C) is dominant when the angular velocity becomes large.

Both the measured and the calculated added mass and added moment of inertia of the rotating plate are shown in Fig. 12A–E. Fig. 12A,B shows the variations of calculated shape factor V_F and calculated position of centre of pressure with distance d. The results for two different plates, plate A and plate B, and two different opening angles, α=0 and α=π/2, are shown. When the opening angle α is zero, the added mass, which is proportional to the shape factor V_F, increases greatly as the distance d becomes small. This is because the fluid between the two plates is pulled by both the plates. The fluid between two plates, then, is more difficult to move than that surrounding an isolated plate, and the resulting flow causes a larger added mass on the plate. Also, it is observed in Fig. 12B that the position of the centre of pressure is not changed when the added mass increases. Therefore, the added moment of inertia increases in proportion to the increase in the added mass. In contrast, the added mass and the position of the centre of pressure are almost constant for the various distances d with α=π/2. The ratio of the measured to the calculated added masses and the ratio of the measured to the calculated added moments of inertia are shown in Fig. 12C and D, respectively. The results indicate good agreement between measurement and calculation. Fig. 12E shows the variation of added mass with the opening angle α for two different distances d. It is observed that the added mass decreases significantly as the opening angle increases, especially in the case of a small distance d. This large variation of the added mass with opening angle gives the considerable values of and in equation 16. The normal force due to the impulsive pressure of the non-circulatory component and each of its components given by equation 16 are shown in Fig. 13 with d=0.02. The chain line indicates the calculated F_N2(NC), where the measured plate movement for d=0.02 is assumed and where the induced velocity generated by the other plate is ignored. The difference between the solid line and the chain line indicates the effect of interference between two plates. This interference increases the normal force due to the impulsive pressure of the non-circulatory component for the period 0<α<0.1 only. The component reduces the total value of the normal force due to the impulsive pressure of the non-circulatory component, and this total value with the interference is smaller than that without the interference during the period α>0.1. In summary, the beating motion, where both the distance d and the opening angle α are small, makes it possible to increase the added mass of a wing. The normal force due to the impulsive pressure of the non-circulatory component, however, is increased during the very limited period in which the opening angle is very small.

A simple model was proposed for analyzing the wing-beating flight of insects (Weis-Fogh, 1973; Ellington, 1984b). This simple model and another simple model newly presented in this paper are compared with the present experimental and numerical calculations in order to make clear the capability of these simple models. The previous simple model by Weis-Fogh or Ellington used a plate element parallel to the rotation axis (plate element 1) as shown in Fig. 2, and the forces acting on this element were considered. The present simple model used a plate element perpendicular to the rotation axis (plate element 2). The calculated results are compared with the experimental results for the single-plate case (i.e. d=∞).

Two typical instants, t=0 and ⁠, are selected. The force and moment actingon a plate are caused by the added mass alone at t=0, and by the ‘quasi-steady’ force alone at ⁠. The word ‘quasi-steady’ means that the forces and the moment are regarded as being proportional to the second power of relative velocity, that is, the second power of angular velocity of the plate.

Fig. 14A–D shows the variations of the non-dimensional shape factors with the variation of the non-dimensional plate geometry ratio R=c_tip/x_tip. The results for the two simple models, obtained by use of equations 23, 24, 31 and 32, are compared with the results obtained by using the present vortex lattice method. It is observed that the previous simple model approach (Weis-Fogh, 1973; Ellington, 1984b) is in good agreement with the vortex lattice method when the ratio R is less than 0.5. In contrast, the present simple model approach provides acceptable solutions when the ratio R is large (R>2). This means that the flow around the plate at the beginning of the plate motion is parallel to plate element 1 for small values of R and to plate element 2 for large values of R near the beginning of the plate motion, t≈0. It is also observed that neither of the simple model approaches provides acceptable predictions for an intermediate value of the ratio R (≈1), which corresponds to that for the so-called ‘low aspect ratio wing’ such as that of a butterfly. In addition, it should be noted that, in using either of the simple models, it is not possible to take into account the interference between the two wings of a pair. Therefore, the present vortex lattice method should be used to analyze precisely a low aspect ratio wing such as a butterfly wing or the interference between two beating wings.

The following characteristics of ‘near fling’ are made clear by the experiments and the numerical calculation. The hydrodynamic force acting on a pair of rotating plates is perpendicular to each of the plate surfaces, and this force is drag because the force is parallel to the inflow velocity due to rotation of the plate. The major part of the hydrodynamic force is composed of two components: that due to the dynamic pressure and that due to the added mass.

The interference between each wing of a pair increases the added mass and the hydrodynamic force due to the dynamic pressure with a small opening angle α and a small distance d between the wings. The hydrodynamic force due to the dynamic pressure is proportional to the bound vortex circulation and to the induced velocity component along the wing surface. The interference between two wings increases the bound vortex circulation near the rotation axis and the induced velocity component along the surface of one wing generated by the vortices shed from the wing and by the bound vortices of the other wing. The increase in the hydrodynamic force due to the interference effect is, however, only important when the opening angle is small. The vertical force, which is balanced with the gravity force during insect flight, is roughly perpendicular to the hydrodynamic force when the opening angle is small. Therefore, the interference effect between a pair of wings is important for insect flight only when the angular velocity or angular acceleration of the flapping angle at a small opening angle is much larger than at a large opening angle.

Three calculation methods, a vortex lattice method and two simple methods, are used to analyze the hydrodynamic force. The newly developed vortex lattice method, in which the plate is represented by the vortex lattice, predicts well every experimental result. This method has the capability of predicting both the interference effect between a pair of wings and the effect of the vortices shed into the flow.

Both the simple methods use a plate element parallel to the rotation axis (plate element 1) or perpendicular to the rotation axis (plate element 2). The simple method using plate element 1 can estimate the added mass of isolated beating wings when the non-dimensional plate geometry ratio R is less than 0.5. This method, therefore, is effective for calculating the added mass of an insect’s wing with a small geometry ratio, such as a dragonfly’s wing, when the interference effect between a pair of wings is negligible. In contrast, the simple method using plate element 2 can estimate the added mass of isolated wings when the geometry ratio is greater than 2. The geometry ratio of butterfly wings is about 1, and the simple method using either element fails to estimate its added mass. In addition, the simple method using either element fails to estimate added mass of any wings when there is an interference effect between a pair of wings.

The quasi-steady force coefficient, which has been used for the simple method using the plate element 1 (Weis-Fogh, 1973; Ellington, 1984b) and is defined by equations 33 and 34, is greater than 3. This is caused by the effects of the vortices shed into the flow. This will make clear the mechanism of generating the extraordinarily large vertical force observed in insect flight (Dudley, 1991).

The above results will be useful for understanding the ‘near fling’ elements of insect flight. The previous simple methods used for analyzing beating wings should be improved to allow a quantitative analysis of ‘near fling’.

 
• [A_n,m]

  matrices of influence coefficients caused by bound vortices on the plate (m⁻¹)

 
• [B_n,m]

  matrices of influence coefficients caused by bound vortices on the other plate (m⁻¹)

 
• b, c

  chord length of plate for three-dimensional test as shown in Fig. 2 (m)

 
• [C_n,4]

  matrix of influence coefficients by vortex elements in wake (m⁻¹)

 
• c_v

  chord length of plate for two-dimensional visualization test (m)

 
• d

  non-dimensional distance between mirror and rotation axis, d₀/x_tip

 
• d₀

  distance between mirror and rotation axis (m)

 
• dm_f

  added mass of two-dimensional plate (kg)

 
• [D_n,1]

  matrix of collocation points (m)

 
• F_N

  normal component of hydrodynamic force acting on plate (N)

 
• F_S

  tangential component of hydrodynamic force acting on plate (N)

 
• F_X, F_Z

  X and Z components of force measured by a load cell (N)

 
• f₁–f₆

  shape factors (m⁴ or m⁵)

 
• g

  acceleration due to gravity (9.81 ms⁻²)

 
• I

  moment of inertia around rotation axis (kg m²)

 
• K

  non-dimensional parameter, b(y)/x_tip

 
• 

  quasi-steady force coefficients at

 
• L

  length between centre axis of load cell and rotation axis (m)

 
• l₁, l₂

  positions of weight (m)

 
• l₃

  height of weight (m)

 
• l_w

  distance between rotation axis and gravitational centre of weight (m)

 
• M_f

  hydrodynamic moment around rotation axis (Nm)

 
• M_Y

  moment around Y-axis measured by a load cell (Nm)

 
• m

  vortex number

 
• m_p

  mass of plate (kg)

 
• m_w

  mass of weight (kg)

 
• p

  angle indicated in Figs 2 and 3 (rad)

 
• R

  non-dimensional parameter, c_tip/x_tip

 
• Re

  Reynolds number

 
• t

  time (s)

 
• t_p

  thickness of plate (m)

 
• t_w

  thickness of weight (m)

 
• u_S, v_S

  x and y components of inflow velocity to bound vortex which is not parallel with x-axis (ms⁻¹)

 
• V_F

  shape factor which is proportional to added mass (m⁴)

 
• V_M

  shape factor which is proportional to added moment of inertia (m⁵)

 
• V_p

  volume of plate (m³)

 
• V_w

  volume of weight (m³)

 
• v_T

  y component of inflow velocity into bound vortex which is parallel with x-axis (ms⁻¹)

 
• X, Y, Z

  earth-fixed coordinate system

 
• x, y, z

  plate-fixed coordinate system

 
• x_C

  x-coordinate at collocation points (m)

 
• x_CG

  x-coordinate of gravitational centre of plate (m)

 
• x_S

  x-coordinate of centre of bound vortex filament which is not parallel with x-axis (m)

 
• x_T

  x-coordinate of centre of bound vortex filament which is parallel with x-axis (m)

 
• α

  opening angle as shown in Fig. 3 (rad)

 
• α₀

  initial opening angle (rad)

 
• Γ

  circulation of bound vortex (m² s⁻¹)

 
• Δl_S

  length of vortex filament which is not parallel to x-axis (m)

 
• Δl_T

  length of vortex filament which is parallel to x-axis (m)

 
• ΔS

  panel area (m²)

 
• Δt

  time step (s)

 
• Δ ϕ

  difference of the non-circulatory component of velocity potential between the upper and lower surface (m² s⁻¹)

 
• 𝒱

  sssssskinematic viscosity (m² s⁻¹)

 
• ρ_f

  density of fluid (kgm⁻³)

 
• ρ_p

  density of plate (kgm⁻³)

 
• ρ_w

  density of weight (kgm⁻³)

 
• ϕ

  velocity potential (m² s⁻¹)

 
• φ

  inclination angle of bound vortex as shown in Fig. 7A (rad)

 
• Ω

  angular velocity in two-dimensional flow visualization test (rad s⁻¹)

SUBSCRIPTS OR SUPERSCRIPTS
 
• C

  circulatory component

 
• f

  value of fluid

 
• i

  i-th time step

 
• i_c

  current time step

 
• j

  vortex number in j-direction as shown in Fig. 7A 

 
• k

  vortex number in k-direction as shown in Fig. 7A 

 
• max

  maximum value

 
• NC

  non-circulatory component

 
• p

  value of plate

 
• tip

  value at tip

 
• w

  value of weight

 
• 1

  normal force or moment due to dynamic pressure

 
• 2

  normal force or moment due to impulsive pressure

It is shown in this section that the exact experimental method (method I) where two plates rotate symmetrically is equivalent to the experimental method used in this paper (method II) where a plate rotates in front of a mirror. Theoretically, both methods will give the same results when the potential flow is assumed. The friction between the flow and the mirror, however, restricts flow along the mirror surface in method II. This creates the possibility of a difference between the results of the two methods.

Fig. 15 shows the arrangement of the experimental apparatus for method I. The force in direction Z, 2F_Z in equation 1, is measured by a load cell. It is predicted that the difference between the two methods is larger when the distance d is smaller and when the rectangular plate (plate A) is used. Therefore, plate A, with the smallest distance in Table 2, d=0.02, is selected for this experiment.

The solid and broken lines in Fig. 16 show time histories of the force in direction Z, F_Z, measured using methods I and II, respectively. They are in good agreement. In addition, the time history of plate movement obtained by method I corresponds closely to the result obtained by method II shown by a short broken line in Fig. 8A. Therefore, it is verified that method I is equivalent to method II in the present research. Method II was used because of the ease with which the experiment could be conducted.