SUMMARY
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.
- thrips
- Thripidae frankliniella
- bristled wing
- membranous wing
- fluid-dynamics
- constant-velocity translation/rotation
- accelerating translation/rotation
Introduction
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.
Materials and methods
Wing shape
The fore- and hindwings of a real thrips Thripidae frankliniella intonsa) 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_{w}c_{m}. The solid wing has a surface area x_{w}c. 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.
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, 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 V_{0}. 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 V_{0} at t=T_{0t}. Because the tank was much larger than the model wings, wall and surface effects can be ignored.
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 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,c} for the two translational motions.
Constant-velocity translation
The forces F_{n}, F_{t}, F_{n,c} and F_{t,c} were measured when they reached constant values. The fluid-dynamic forces acting on the wing only were calculated using the expressions, F_{n}—F_{n,c} and F_{t}—F_{t,c}. The lift coefficient C_{L} and drag coefficient C_{D} were obtained by non-dimensionalizing the measured fluid-dynamic forces as follows: 2 3
Accelerating translation
The forces F_{n}, F_{t}, F_{n,c} and F_{t,c} were measured at 0≤t≤T_{0t}. F_{n} and F_{t} 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. F_{n,c} and F_{t,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, F_{n}—F_{n,c} and F_{t}—F_{t,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 F_{n}—F_{n,c}—m_{w}Ẍsinα and F_{t}—F_{t,c}—m_{w}Ẍcosα, respectively, where m_{w} is the mass of the wing, and m_{w}Ẍsinα and m_{w}Ẍcosα are the normal and tangential components, respectively, of the inertial force acting on the wing. C_{L} and C_{D} 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 (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_{z} and 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°.
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, T_{0r} and Φ/2 are the period and amplitude, respectively, of accelerating rotation, and (π/2√2T_{0r})(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. C_{L} and C_{D} 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≤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}.
C_{L} and C_{D} 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.
Results
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 T acting 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_{D} vary 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 V_{0}. Furthermore, C_{L} and C_{D} were larger for t≤T_{0t} than 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<10^{3}, C_{L} and C_{D}, which are non-dimensionalized by Ẋ^{2}, 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_{D} were 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.
Discussion
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_{w}f)^{2} for 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_{w}f)^{2}, where x_{w}f is proportional to the mean velocity of the flow around the wing. Therefore, the parameter mg/S_{tot}(x_{w}f)^{2} reflects the coefficient of vertical fluid-dynamic force generated by an insect. For the thrips mg/S_{tot}(x_{w}f)^{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_{w}f)^{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 5×10^{7} 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 (6×10^{7} 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
- V_{0}
- 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
ACKNOWLEDGEMENTS
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.
- © The Company of Biologists Limited 2002