Thrips fly at a chord-based Reynolds number of approximately 10 using bristled rather than solid wings. We tested two dynamically scaled mechanical models of a thrips forewing. In the bristled design, cylindrical rods model the bristles of the forewing; the solid design was identical to the bristled one in shape, but the spaces between the `bristles' were filled in by membrane. We studied four different motion patterns: (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. Fluid-dynamic forces acting on the bristled model wing were a little smaller than those on the solid wing. Therefore, the bristled wing of a thrips cannot be explained in terms of increased fluid-dynamic forces.

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.

Wing shape

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 xw is wing length, c is chord length, cm is membrane width, d is cylinder diameter, ch1 is the length of cylinders on the leading edge of the membrane, ch2 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(ch1+ch2) and the membrane area xwcm. The solid wing has a surface area xwc. The parameters xw/c, cm/c, d/xw, ch1/c, ch2/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. 1.

Test wings. (A) Photomicrograph of a thrip's wings. (B,C) Diagrams of a bristled model wing (B) and a solid model wing (C). c, chord length of a wing; cm, membrane width of a bristled wing; ch1, length of cylinders on the leading edge; ch2, length of cylinders on the trailing edge; D,distance between neighbouring cylinders or bristles; d, diameter of cylinders or bristles; tm, thickness of a wing; xw, wing length; x,y,z, wing-fixed coordinate system.

Fig. 1.

Test wings. (A) Photomicrograph of a thrip's wings. (B,C) Diagrams of a bristled model wing (B) and a solid model wing (C). c, chord length of a wing; cm, membrane width of a bristled wing; ch1, length of cylinders on the leading edge; ch2, length of cylinders on the trailing edge; D,distance between neighbouring cylinders or bristles; d, diameter of cylinders or bristles; tm, thickness of a wing; xw, wing length; x,y,z, wing-fixed coordinate system.

Table 1.

Size of model wings and thrips' wing

Wingc (mm)xw/cd/ccm/cch1/cch2/ctm/cS (mm2)D/dn
Thrips* 3×10-1 2.7 5×10-3 2×10-1 1.3×10-1 6.7×10-1 6×10-2 10 50 
Bristled model 60 3.0 5×10-3 2×10-1 1.8×10-1 6.2×10-1 10-2 2.9×103 10 51 
Solid model 60 3.0 10-2 1.1×104 
Wingc (mm)xw/cd/ccm/cch1/cch2/ctm/cS (mm2)D/dn
Thrips* 3×10-1 2.7 5×10-3 2×10-1 1.3×10-1 6.7×10-1 6×10-2 10 50 
Bristled model 60 3.0 5×10-3 2×10-1 1.8×10-1 6.2×10-1 10-2 2.9×103 10 51 
Solid model 60 3.0 10-2 1.1×104 

c, chord length; xw, wing length; d,diameter of bristles or cylinders; cm, membrane width; ch1, length of bristles or cylinders on the leading edge; ch2, length of bristles or cylinders on the trailing edge; tm, thickness of the wing; S, wing surface area; D, distance between neighbouring bristles or cylinders; n,the number of bristles or cylinders.

*

Data for the thrips' wing were obtained by Tanaka(1995).

Forward motion

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, LX=800 mm, LY=400 mm and LZ=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, Fx and Fz, and the moment around the y axis, My. The maximum load for Fx and Fz was 1 N and that for My was 0.01 N m.

Fig. 2.

Experimental apparatus used to measure the fluid-dynamic forces of a wing in forward motion (constant-velocity translation and accelerating translation). X, Y, Z, earth-fixed coorinate system; x, y,z, wing-fixed coordinate system; LX, LY, LZ, dimensions of the tank;α, angle of attack.

Fig. 2.

Experimental apparatus used to measure the fluid-dynamic forces of a wing in forward motion (constant-velocity translation and accelerating translation). X, Y, Z, earth-fixed coorinate system; x, y,z, wing-fixed coordinate system; LX, LY, LZ, dimensions of the tank;α, angle of attack.

The cross talk between Fx, Fz and My was small, and the measured forces Fx and Fz were considered to be equal to the normal and tangential forces, Fn and Ft, respectively, on the wing(Fx=Fn, Fz=-Ft).

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 tT0t(tT0t =4 or 10 s), where t is time and T0t is the period of accelerated motion. The forward velocity reached a terminal value V0 at t=T0t. Because the tank was much larger than the model wings, wall and surface effects can be ignored.

Table 2.

Wing motions

Table 2.

Wing motions

Table 2 also shows the Reynolds number Re calculated as follows:
1
where ν is the kinematic viscosity of the liquid. During constant-velocity translation and accelerating translation, Re=12, which is similar to the Re=10 for a flying thrips(Tanaka, 1995).

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

Constant-velocity translation

The forces Fn, Ft, Fn,c and Ft,c were measured when they reached constant values. The fluid-dynamic forces acting on the wing only were calculated using the expressions, FnFn,c and FtFt,c. The lift coefficient CL and drag coefficient CD were obtained by non-dimensionalizing the measured fluid-dynamic forces as follows:
2
3

Accelerating translation

The forces Fn, Ft, Fn,c and Ft,c were measured at 0≤tT0t. Fn and Ft are the sum of the fluid-dynamic and inertial forces acting on the joint cylinder, the fluid-dynamic and inertial forces acting on the wing and the inertial forces on the load cell. Fn,cand Ft,c are the sum of the fluid-dynamic and inertial forces acting on the joint cylinder and the inertial forces acting on the load cell. The load cell measured an inertial force proportional to the accelerated mass attached to the strain gauge in the load cell. Therefore, FnFn,c and FtFt,c are the sum of the fluid-dynamic and inertial forces acting on the wing. The normal and tangential fluid-dynamic forces acting only on the wing are given by FnFn,cmwsinαand FtFt,cmwcosα,respectively, where mw is the mass of the wing, and mwsinα and mwcosα are the normal and tangential components, respectively, of the inertial force acting on the wing. CL and CD were obtained by non-dimensionalizing the measured fluid-dynamic forces as follows:
4
5

Rotational motion

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(LX, LY=500 mm and LZ=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, Fz, and the moment around the Z axis, Mz. The maximum load was 5 N for Fz and 0.25 N m for Mz. 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, Fz and Mz are affected by all the forces and moments acting on the load cell. However,because the cross talk between Fz and Mz was small, measured values of Fzand Mz 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.

Fig. 3.

Experimental apparatus used to measure the fluid-dynamic forces of a wing in rotational motion (constant-velocity rotation and accelerating rotation). The measuring system (MS) comprises a load cell and a motor. X, Y, Z,earth-fixed coordinate system; x, y, z, wing-fixed coordinate system; LX, LY, LZ,dimensions of the tank; LZ1, depth of liquid;0.5LX, 0.5LY,0.5LZ, position of a root of a wing.

Fig. 3.

Experimental apparatus used to measure the fluid-dynamic forces of a wing in rotational motion (constant-velocity rotation and accelerating rotation). The measuring system (MS) comprises a load cell and a motor. X, Y, Z,earth-fixed coordinate system; x, y, z, wing-fixed coordinate system; LX, LY, LZ,dimensions of the tank; LZ1, depth of liquid;0.5LX, 0.5LY,0.5LZ, position of a root of a wing.

The tank was filled to the depth LZ1 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 0.7LZ1. 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°.

Table 2 lists the rotational angle ϕ during constant-velocity rotation and accelerating rotation and lists Re defined for constant-velocity rotation as
6
and for accelerating rotation,
7
where ω is the rotational angular velocity for constant-velocity rotation, T0r and Φ/2 are the period and amplitude,respectively, of accelerating rotation, and(π/2√2T0r)(Φ/2) is the averaged angular velocity. For constant-velocity rotation and accelerating rotation, Re was approximately 10, which is close to the Re for a flying thrips (Tanaka,1995).

Constant-velocity rotation

We measured the force in the -Z direction, i.e. thrust T,and the moment around the Z axis, i.e. torque Q, after the wing had completed 30 rotations. The measured T and Q were considered to be equal to the fluid-dynamic thrust and torque of the wing because the forces acting on the joint cylinder were much smaller than those acting on the wing. CL and CD were determined by non-dimensionalizing the measured fluid-dynamic thrust and torque as follows (Ellington,1984):
8
9
where x is span-wise axis, shown in Fig. 1.

Accelerating rotation

Thrust T and torque Q were measured for 0≤t≤T0r 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, Qc, 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—Qc.

CL and CD were determined by non-dimensionalizing the measured fluid-dynamic thrust and torque as follows(Ellington, 1984):
10
and
11
where ϕ is the instantaneous angular velocity.

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. 4.

Ratio of forces (lift L, drag D in constant-velocity translation and thrust T in constant-velocity rotation) or torque(Q in constant-velocity rotation) acting on the bristled wing to those on the solid wing. Open squares, L, filled squares, D,in constant-velocity translation; open circles, T, filled circles, Q, in constant-velocity rotation.

Fig. 4.

Ratio of forces (lift L, drag D in constant-velocity translation and thrust T in constant-velocity rotation) or torque(Q in constant-velocity rotation) acting on the bristled wing to those on the solid wing. Open squares, L, filled squares, D,in constant-velocity translation; open circles, T, filled circles, Q, in constant-velocity rotation.

Fig. 5 shows CL and CD plotted versusα for constant-velocity translation and constant-velocity rotation. For both motions, CL and CD of the bristled wing were larger than those of the solid wing. The differences in CL and CD 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 T0t=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 CL and CDvary with distance travelled for accelerating translation when α=45° and T0t=4 s. When t=T0t, the non-dimensional displacement X/c was approximately 0.8. The figure shows that neither CL nor CD reached a constant value when t=T0t and that the forward velocity reached its terminal value V0. Furthermore, CL and CD were larger for tT0tthan at t=T0t. Fluid-dynamic forces due to added mass (Ellington, 1984), which act on the wings when t<T0t, are negligible. The larger values of CL and CD for tT0t might be explained in two ways. First, Re defined by instantaneous forward velocity was smaller for t<T0t than at t=T0t. For Re<103, CL and CD, which are non-dimensionalized by 2,increase as Re decreases. For Re<1, CL and CD are proportional to 1/Re and ReCL and ReCD are independent of Re (e.g. Hoerner,1965). Second, wing motion accelerated while tT0t, and this acceleration caused an increase in CL and CD. This increase is expected to be caused by `delayed stall'(Dickinson et al., 1999).

Fig. 5.

Lift coefficient CL (A) and drag coefficient CD (B) for steady motion (constant-velocity translation and constant-velocity rotation). Open circles, the bristled wing in constant-velocity translation; filled circles, the bristled wing in constant-velocity rotation; open squares, the solid wing in constant-velocity translation; filled squares, the solid wing in constant-velocity rotation.

Fig. 5.

Lift coefficient CL (A) and drag coefficient CD (B) for steady motion (constant-velocity translation and constant-velocity rotation). Open circles, the bristled wing in constant-velocity translation; filled circles, the bristled wing in constant-velocity rotation; open squares, the solid wing in constant-velocity translation; filled squares, the solid wing in constant-velocity rotation.

Fig. 6.

Ratios of lift (open circles) and drag (filled circles) acting on the bristled wing in accelerating translation (T0t=4 s andα=45 °) to those on the solid wing. t, time (s); T0t, period of acceleration phase in translational motion;α, angle of attack.

Fig. 6.

Ratios of lift (open circles) and drag (filled circles) acting on the bristled wing in accelerating translation (T0t=4 s andα=45 °) to those on the solid wing. t, time (s); T0t, period of acceleration phase in translational motion;α, angle of attack.

Fig. 7.

Changes in time of CL and CD (A)and those of ReCL and ReCD (B) for accelerating translation, when T0t=4 s and α=45°. t, time (s); T0t, period of acceleration phase in translational motion; α, angle of attack. Open circles, CL or ReCL for the bristled wing;filled circles, CD or ReCD for the bristled wing; open squares, CL or ReCL for the solid wing; filled squares, CD or ReCD for the solid wing.

Fig. 7.

Changes in time of CL and CD (A)and those of ReCL and ReCD (B) for accelerating translation, when T0t=4 s and α=45°. t, time (s); T0t, period of acceleration phase in translational motion; α, angle of attack. Open circles, CL or ReCL for the bristled wing;filled circles, CD or ReCD for the bristled wing; open squares, CL or ReCL for the solid wing; filled squares, CD or ReCD for the solid wing.

To test the first hypothesis, we looked at how ReCL and ReCD changed over time for accelerating translation whenα=45 ° and T0t=4 s(Fig. 7B). These changes over time were smaller than those of CL and CD shown in Fig. 7A. However, ReCL and ReCDwere larger for t<T0t than at t=T0t. Therefore, the second hypothesis is also needed to explain the differences in CL and CD for t<T0t than at t=T0t. This might apply not just for T0t=4 s but also for T0t=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. 8.

Ratios of thrust T and torque Q acting on the bristled wing in accelerating rotation (α=20 ° and α=45 °) to those on the solid wing. t, time (s); T0r, period of acceleration phase in rotational motion; α, angle of attack. Open circles, T at α=20 °; filled circles, Q atα=20 °; open squares, T at α=45 °; filled squares, Q at α=45 °.

Fig. 8.

Ratios of thrust T and torque Q acting on the bristled wing in accelerating rotation (α=20 ° and α=45 °) to those on the solid wing. t, time (s); T0r, period of acceleration phase in rotational motion; α, angle of attack. Open circles, T at α=20 °; filled circles, Q atα=20 °; open squares, T at α=45 °; filled squares, Q at α=45 °.

Fig. 9A shows changes over time of CL and CD for accelerating rotation when α=45 °. The coefficients CL and CD of the solid wing were smaller than those of the bristled wing. The CL and CD for t<T0r were larger than those at t=T0r. The fluid-dynamic forces due to added mass(Ellington, 1984) are negligible while t<T0r. The changes over time of ReCL and ReCD in Fig. 9B show the differences in CL and CD for t<T0r and t=T0r. Just as during accelerating translation, ReCL and ReCD are larger for t<T0r,and again this difference is due to delayed stall.

Fig. 9.

Changes in time of CL and CD (A)and ReCL and ReCD (B) for accelerating rotation, when α=45°. t, time (s); T0r,period of accelerated phase in rotational motion; α, angle of attack. Open circles, CL or ReCL for the bristled wing; filled circles, CD or ReCD for the bristled wing; open squares, CL or ReCL for the solid wing; filled squares, CD or ReCD for the solid wing.

Fig. 9.

Changes in time of CL and CD (A)and ReCL and ReCD (B) for accelerating rotation, when α=45°. t, time (s); T0r,period of accelerated phase in rotational motion; α, angle of attack. Open circles, CL or ReCL for the bristled wing; filled circles, CD or ReCD for the bristled wing; open squares, CL or ReCL for the solid wing; filled squares, CD or ReCD for the solid wing.

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/Stot(xwf)2for the insects listed in Fig. 10A, where g is the acceleration due to gravity, Stot is the total wing surface area of four wings of an insect, and xw is the length of the forewing. The fluid-dynamic force generated by a wing is proportional to Stot(xwf)2, where xwf is proportional to the mean velocity of the flow around the wing. Therefore, the parameter mg/Stot(xwf)2reflects the coefficient of vertical fluid-dynamic force generated by an insect. For the thrips mg/Stot(xwf)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/Stot(xwf)2≈25 for thrips can be explained by the larger values of CL and CD for a bristled model wing compared with the coefficients for the solid model wing.

Fig. 10.

Comparison of flight data for a thrip and for other insects. (A) mass (kg) m versus wing-beat frequency (Hz) f, (B) mass (kg) m versus a parameter indicating vertical force coefficient mg/Stot(xwf)2. Data for the thrips (filled circle) are from Tanaka(1995). Data for the following insects (open circles) are from Azuma(1992): 1Bemisia tabaci, 2Aleurothrixus floccosus, 3Aphis gossypii, 4Acyrthosiphon kondoi, 5Aedes nearcticus, 6Musca domestica, 7Panorpa communis L., 8Pyrosoma minimum Harr., 9Amonophila sabulosa V.del, 10Sarcophaga carnaria L., 11Volucella pellucens Meig., 12Apis mellifica L., 13Telepharus fuscus, 14Calopteryx splendes Harr., 15Pieris brassica L., 16Vanessa atolanta L., 17Plusia gamma L., 18Talanus affioris, 19Vespa germanica, 20Orthetrum caerulescens Fabr., 21Tabanus botinus, 22Papilio podalirius, 23Macroglossa stellatorum L., 24Bombus terrestris Fabr., 25Aeschna mixtra Latr., 26Cetonia aurata, 27Brachytron pratense Mull., 28Vespa crabro L., 29Xylocope violacea, 30Anax parthenope, 31Melolontha vulgaris Fabr., 32Schistocerca gregaria, 33Lucanus corcus.

Fig. 10.

Comparison of flight data for a thrip and for other insects. (A) mass (kg) m versus wing-beat frequency (Hz) f, (B) mass (kg) m versus a parameter indicating vertical force coefficient mg/Stot(xwf)2. Data for the thrips (filled circle) are from Tanaka(1995). Data for the following insects (open circles) are from Azuma(1992): 1Bemisia tabaci, 2Aleurothrixus floccosus, 3Aphis gossypii, 4Acyrthosiphon kondoi, 5Aedes nearcticus, 6Musca domestica, 7Panorpa communis L., 8Pyrosoma minimum Harr., 9Amonophila sabulosa V.del, 10Sarcophaga carnaria L., 11Volucella pellucens Meig., 12Apis mellifica L., 13Telepharus fuscus, 14Calopteryx splendes Harr., 15Pieris brassica L., 16Vanessa atolanta L., 17Plusia gamma L., 18Talanus affioris, 19Vespa germanica, 20Orthetrum caerulescens Fabr., 21Tabanus botinus, 22Papilio podalirius, 23Macroglossa stellatorum L., 24Bombus terrestris Fabr., 25Aeschna mixtra Latr., 26Cetonia aurata, 27Brachytron pratense Mull., 28Vespa crabro L., 29Xylocope violacea, 30Anax parthenope, 31Melolontha vulgaris Fabr., 32Schistocerca gregaria, 33Lucanus corcus.

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

  •  
  • Cm

    Membrane width of a bristled wing

  •  
  • Ch1

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

  •  
  • Ch2

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

  •  
  • CL, CD

    Lift and drag coefficients, respectively

  •  
  • d

    Diameter of cylinders or bristles

  •  
  • D

    Distance between neighbouring cylinders or bristles

  •  
  • f

    Wing-beat frequency

  •  
  • Fn, Ft

    Normal and tangential forces, respectively

  •  
  • Fn,c, Ft,c

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

  •  
  • FZ, MZ

    Force in Z axis and moment around Z axis,respectively

  •  
  • Fx, Fz, My

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

  •  
  • g

    Acceleration of gravity

  •  
  • LX, LY, LZ

    Dimensions of the tank

  •  
  • LZ1

    Depth of liquid

  •  
  • m

    Mass of an insect

  •  
  • mw

    Mass of a wing

  •  
  • n

    The number of cylinders or bristles of a bristled wing

  •  
  • Q

    Torque

  •  
  • Qc

    Torque measured in air

  •  
  • Re

    Reynolds number

  •  
  • S

    Wing surface area

  •  
  • Stot

    Total wing surface area of an insect

  •  
  • t

    Time

  •  
  • tm

    Thickness of a wing

  •  
  • T

    Thrust

  •  
  • T0r

    Period of accelerated phase in rotational motion

  •  
  • T0t

    Period of accelerated phase in translational motion

  •  
  • V0

    Terminal forward velocity

  •  
  • x, y, z

    Wing-fixed coordinate system

  •  
  • xw

    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.

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