Fluid-dynamic characteristics of a bristled wing

Numerous animals use hairy appendages for feeding and locomotion. Extensive research has been carried out on the flow volume through the hairs on feeding appendages (e.g. Koehl, 1983, 1995; Hansen and Tiselius, 1992), but there is not much information on the fluid-dynamic forces generated by hairy appendages (Horridge, 1956; Ellington, 1980; Cheer and Koehl, 1987; Kuethe, 1975; Tanaka, 1995). How to estimate the fluid-dynamic forces acting on the hairy appendages at low and very low Reynolds number has not therefore been clarified.

To estimate the fluid-dynamic forces acting on the hairy appendages and understand the fluid-dynamic mechanisms of thrips flight, we measured the fluid-dynamic characteristics of a dynamically scaled model of the forewing. Four different wing motions were studied: forward motion at a constant velocity (constant-velocity translation), forward motion at a translational acceleration (accelerating translation), rotational motion at a constant angular velocity (constant-velocity rotation) and rotational motion at an angular acceleration (accelerating rotation). For comparison, the fluid-dynamic characteristics of a solid model wing were also measured. The solid wing had the same outline as the bristled model, but was made from a solid flat plate of the same thickness as the bristle diameter. Comparing the fluid-dynamic performance of the bristled and the solid wing might help to clarify why a small insect, such as a thrips, uses bristled wings for flight.

The fore- and hindwings of a real thrips Thripidae frankliniellaintonsa) are shown in Fig. 1A. Fig. 1B,C shows diagrams of the bristled wing and solid model wings, respectively. The bristled wing comprises a membrane and 51 cylinders. Table 1 lists the dimensions of a thrips' forewing, measured by Tanaka(1995), and of the model wings, where x_w is wing length, c is chord length, c_m is membrane width, d is cylinder diameter, c_h1 is the length of cylinders on the leading edge of the membrane, c_h2 is the length of cylinders on the trailing edge of the membrane, n is the number of cylinders on each edges, D is the distance between neighbouring cylinders, and S is the wing surface area. S for the bristled wing is the sum of the cylinder frontal area nd(c_h1+c_h2) and the membrane area x_wc_m. The solid wing has a surface area x_wc. The parameters x_w/c, c_m/c, d/x_w, c_h1/c, c_h2/c, n and D/d are similar for both the bristled wing and a real thrips' forewing, maintaining geometric scaling. The fluid-dynamic characteristics of the bristled wing and the solid wing were measured for four motion patterns.

Fig. 2 shows the apparatus used to measure the fluid-dynamic forces acting on the model wings in forward motion. A tank (dimensions in X, Y and Z directions, L_X=800 mm, L_Y=400 mm and L_Z=500 mm, respectively) was filled with an aqueous solution of glycerine. The wing was suspended from a load cell (LMC3729-1N,Nissho Electric Works, Japan) via an 8 mm diameter joint cylinder. The load cell can measure forces in the x and z directions, F_x and F_z, and the moment around the y axis, M_y. The maximum load for F_x and F_z was 1 N and that for M_y was 0.01 N m.

The cross talk between F_x, F_z and M_y was small, and the measured forces F_x and F_z were considered to be equal to the normal and tangential forces, F_n and F_t, respectively, on the wing(F_x=F_n, F_z=-F_t).

The wing was moved in the X direction at a constant angle of attack α between -10° and 45° as described for constant-velocity translation and accelerating translation in Table 2. During constant-velocity translation, the wing moved at a constant forward velocity V₀. During accelerating translation, the wing underwent sinusoidal acceleration for t≤T_0t(t≤T_0t =4 or 10 s), where t is time and T_0t is the period of accelerated motion. The forward velocity reached a terminal value V₀ at t=T_0t. Because the tank was much larger than the model wings, wall and surface effects can be ignored.

The fluid-dynamic forces acting on the wing were measured as follows. First, the normal and tangential forces F_n,c and F_t,c were measured for the wing mount only without the wing connected to the joint cylinder. Next, we measured the normal and tangential forces F_n and F_t generated by both the wing and its mount. The fluid-dynamic forces acting on the wing only were calculated from the measured forces, F_n, F_t, F_n,c and F_t,cfor the two translational motions.

Fig. 3 shows the apparatus used to measure the fluid-dynamic forces acting on the model wings in rotational motion. The model wing was suspended in a tank(L_X, L_Y=500 mm and L_Z=1000 mm) filled with an aqueous solution of glycerine. The wing was mounted onto a load cell (LMC2909, Nissho Electric Works, Japan)and a motor via a 6 mm diameter joint cylinder. The wing rotated around the joint cylinder in the X—Y plane. The load cell measured force in the Z direction, F_z, and the moment around the Z axis, M_z. The maximum load was 5 N for F_z and 0.25 N m for M_z. When forces in the X, Y and Z directions and moments around the X, Y and Z axes act on the load cell, the output signal from the load cell, F_z and M_z are affected by all the forces and moments acting on the load cell. However,because the cross talk between F_z and M_z was small, measured values of F_zand M_z were considered to be equal to the force in the Z direction and the moment around the Z axis actually acting on the load cell, respectively.

The tank was filled to the depth L_Z1 of 980 mm with an aqueous solution of glycerine. The rotational axis of the wing was at the centre of the tank in the X—Y plane. The distance between the rotational plane and the bottom of the tank was 0.7L_Z1. All tank dimensions are large enough for surface and wall effects to be negligible.

The geometrical angle of attack α was defined as the angle between the Z axis and a vector normal to the wing. The angle of attackα was set between -10° and 45°.

Thrust T and torque Q were measured for 0≤t≤T_0r in an aqueous solution of glycerine. As in the case of constant-velocity rotation, we neglected the forces acting on the joint cylinder and assumed that the measured thrust T is equal to the fluid-dynamic thrust. The measured torque is the sum of the fluid-dynamic torque acting on both the wing and the joint cylinder, as well as the inertial torque acting on the wing, on the joint cylinder and on the load cell. Because the fluid-dynamic torque acting on the joint cylinder is much smaller than that acting on the wing, the former torque can be neglected. We measured the torque in air to estimate the inertial torque acting on the wing, on the joint cylinder and on the load cell. The measured torque in air, Q_c, was approximately equal to the inertial torque acting on the wing, on the joint cylinder and on the load cell because their density is much larger than the density of air and, hence, the fluid-dynamic torque in air was much smaller than the inertial torque acting on these three components(wing, joint cylinder and load cell). Therefore, the fluid-dynamic torque acting on the wing was obtained from Q—Q_c.

Fig. 4 compares the fluid-dynamic forces for the steady motions, constant-velocity translation and constant-velocity rotation, and shows the ratio of lift L acting on the bristled wing to that acting on the solid wing and the ratio of drag D acting on the bristled wing to that acting on the solid wing for constant-velocity translation. Also shown is the ratio of thrust Tacting on the bristled wing to that acting on the solid wing and the ratio of torque Q acting on the bristled wing to that acting on the solid wing, for constant-velocity rotation. The ratios of lift, drag, thrust and torque acting on the bristled wing to those on the solid wing were a little less than 1, except for constant-velocity rotation (T; at α=10° and 20 °). During steady motion (constant-velocity translation and constant-velocity rotation), the fluid-dynamic forces acting on the bristled wing were smaller than those acting on the solid wing, except for constant-velocity rotation (T; at α=10 ° and 20 °).

Fig. 5 shows C_L and C_D plotted versusα for constant-velocity translation and constant-velocity rotation. For both motions, C_L and C_D of the bristled wing were larger than those of the solid wing. The differences in C_L and C_D between constant-velocity rotation and constant-velocity translation for the bristled wing are larger than those for the solid wing. Hence, the flow around the bristled wing should exhibit large differences between constant-velocity rotation and constant-velocity translation than the solid wing. Fig. 6 shows how lift and drag change with distance travelled for the solid wing during accelerating translation (α=45 ° and T_0t=4 s). The lift-to-drag ratio was between 0.8 and 1. During this unsteady translation,the fluid-dynamic forces acting on the bristled wing were smaller than those acting on the solid wing. Fig. 7A shows how C_L and C_Dvary with distance travelled for accelerating translation when α=45° and T_0t=4 s. When t=T_0t, the non-dimensional displacement X/c was approximately 0.8. The figure shows that neither C_L nor C_D reached a constant value when t=T_0t and that the forward velocity Ẋ reached its terminal value V₀. Furthermore, C_L and C_D were larger for t≤T_0tthan at t=T_0t. Fluid-dynamic forces due to added mass (Ellington, 1984), which act on the wings when t<T_0t, are negligible. The larger values of C_L and C_D for t≤T_0t might be explained in two ways. First, Re defined by instantaneous forward velocity Ẋ was smaller for t<T_0t than at t=T_0t. For Re<10³, C_L and C_D, which are non-dimensionalized by Ẋ²,increase as Re decreases. For Re<1, C_L and C_D are proportional to 1/Re and ReC_L and ReC_D are independent of Re (e.g. Hoerner,1965). Second, wing motion accelerated while t≤T_0t, and this acceleration caused an increase in C_L and C_D. This increase is expected to be caused by `delayed stall'(Dickinson et al., 1999).

To test the first hypothesis, we looked at how ReC_L and ReC_D changed over time for accelerating translation whenα=45 ° and T_0t=4 s(Fig. 7B). These changes over time were smaller than those of C_L and C_D shown in Fig. 7A. However, ReC_L and ReC_Dwere larger for t<T_0t than at t=T_0t. Therefore, the second hypothesis is also needed to explain the differences in C_L and C_D for t<T_0t than at t=T_0t. This might apply not just for T_0t=4 s but also for T_0t=10 s.

Fig. 8 shows the changes over time of the ratios of thrust T and torque Q acting on the bristled wing to those acting on the solid wing for accelerating rotation for α=20 ° and 45 °. These ratios were less than 1, except for the ratio at α=20 °, when the fluid-dynamic forces acting on the bristled wing were larger than those acting on the solid wing.

Fig. 9A shows changes over time of C_L and C_D for accelerating rotation when α=45 °. The coefficients C_L and C_D of the solid wing were smaller than those of the bristled wing. The C_L and C_D for t<T_0r were larger than those at t=T_0r. The fluid-dynamic forces due to added mass(Ellington, 1984) are negligible while t<T_0r. The changes over time of ReC_L and ReC_D in Fig. 9B show the differences in C_L and C_D for t<T_0r and t=T_0r. Just as during accelerating translation, ReC_L and ReC_D are larger for t<T_0r,and again this difference is due to delayed stall.

Fluid-dynamic forces acting on the geometrically scaled bristled model wing were smaller than those acting on the solid wing. With a few exceptions, this result was valid for all the four wing motions: (i) forward motion at a constant forward velocity, (ii) forward motion at a translational acceleration, (iii) rotational motion at a constant angular velocity and (iv)rotational motion at an angular acceleration. The bristled wings of a thrips cannot therefore be explained by an augumentation of fluid-dynamic performance.

Fig. 10A shows the relationship between mass m and wing-beat frequency f for a thrips (Tanaka, 1995) and a variety of other insects (Azuma,1992). The wing-beat frequency f of the thrips is 200 Hz,which is relatively low for its body mass(m≈6×10^-8 kg) compared with larger insects, but similar to that of other small insects, such as Bemisia tabaci,Aleurothrixus floccosus, Aphis gossypii and Acyrthosiphon kondoi(numbered 1-4 in Fig. 10A,respectively). Fig. 10B shows the values of mg/S_tot(x_wf)²for the insects listed in Fig. 10A, where g is the acceleration due to gravity, S_tot is the total wing surface area of four wings of an insect, and x_w is the length of the forewing. The fluid-dynamic force generated by a wing is proportional to S_tot(x_wf)², where x_wf is proportional to the mean velocity of the flow around the wing. Therefore, the parameter mg/S_tot(x_wf)²reflects the coefficient of vertical fluid-dynamic force generated by an insect. For the thrips mg/S_tot(x_wf)²≈25;this is larger than that for Bemisia tabaci, Aleurothrixus floccosus,Aphis gossypii and Acyrthosiphon kondoi, which have membranous wings. The larger value of mg/S_tot(x_wf)²≈25 for thrips can be explained by the larger values of C_L and C_D for a bristled model wing compared with the coefficients for the solid model wing.

The resultant force of the lift and drag generated by the thrips was approximately 5×10⁷ N at any flapping angle with the following assumptions: (i) the thrips has four wings whose size is shown in Table 1; (ii) the flapping motion is the same as defined for the accelerating rotation at f=200 Hz; and (iii) the geometrical angle of attack is 45°, and changes over time in the lift and drag coefficients are given by those of the bristled wing in Fig. 9A. The estimated value of the vector sum of lift and drag is close to the gravitational force acting on the thrips (6×10⁷ N). However, to understand more fully the flight of the thrips, we need a more precise estimate of the fluid-dynamic forces generated by their wings, based on more accurate data on wing morphology and kinematics, on the variation in angle of attack(feathering angle), and lift and drag coefficients measured over several consecutive wing beats.

List of symbols
 
• C

  Chord length of a wing

 
• C_m

  Membrane width of a bristled wing

 
• C_h1

  Length of cylinders or bristles attached at the leading edge of the bristled wing

 
• C_h2

  Length of cylinders or bristles attached at the trailing edge of the bristled wing

 
• C_L, C_D

  Lift and drag coefficients, respectively

 
• d

  Diameter of cylinders or bristles

 
• D

  Distance between neighbouring cylinders or bristles

 
• f

  Wing-beat frequency

 
• F_n, F_t

  Normal and tangential forces, respectively

 
• F_n,c, F_t,c

  Normal and tangential forces, respectively, measured on the wing mount without a wing connected to it

 
• F_Z, M_Z

  Force in Z axis and moment around Z axis,respectively

 
• F_x, F_z, M_y

  Forces in x,z axes and moment around y axis,respectively

 
• g

  Acceleration of gravity

 
• L_X, L_Y, L_Z

  Dimensions of the tank

 
• L_Z1

  Depth of liquid

 
• m

  Mass of an insect

 
• m_w

  Mass of a wing

 
• n

  The number of cylinders or bristles of a bristled wing

 
• Q

  Torque

 
• Q_c

  Torque measured in air

 
• Re

  Reynolds number

 
• S

  Wing surface area

 
• S_tot

  Total wing surface area of an insect

 
• t

  Time

 
• t_m

  Thickness of a wing

 
• T

  Thrust

 
• T_0r

  Period of accelerated phase in rotational motion

 
• T_0t

  Period of accelerated phase in translational motion

 
• V₀

  Terminal forward velocity

 
• x, y, z

  Wing-fixed coordinate system

 
• x_w

  Wing length

 
• X, Y, Z

  Earth-fixed coordinate system

 
• α

  Angle of attack

 
• ν

  Kinematic viscosity

 
• ρ

  Density of fluid

 
• Φ/2

  Amplitude of accelerating rotational motion

 
• Φ

  Rotational angle

 
• ω

  Angular velocity for constant-velocity rotation

We would like to thank the anonymous referees for their critical and important comments on the manuscript. This research was financially supported by the Research and Development for Applied Advanced Computational Science and Technology of the Japan Science and Technology Corporation.