J. exp. Biol. 83, 27 248 (993) Printed in Great Britain The Company of Biologists Limited 993 27 FUNDAMENTAL ANALYSIS OF THREE-DIMENSIONAL NEAR FLING SHIGERU SUNADA Exploratory Research for Advanced Technology, Tokyo, Japan KEIJI KAWACHI, ISAO WATANABE AND AKIRA AZUMA Research Center for Advanced Science and Technology, University of Tokyo, Tokyo, Japan Accepted 8 June 993 Summary A series of experiments on three-dimensional near fling was carried out. Two pairs of plates, rectangular and triangular, were selected, and the distance between the rotation axes of the two plates of each pair was varied. The motion of the plates as well as the forces and the moment were measured, and the interference between the two plates of a pair was studied. In addition, a method of numerical calculation was developed to aid in the understanding of the experimental results. The interference between the two plates of a pair, which acted to increase both the added mass of each plate and the hydrodynamic force due to dynamic pressure, was noted only when the opening angle between the plates was small. The hydrodynamic forces were strongly influenced by separated vortices that occurred during the rotation. A method of numerical calculation, which took into account the effect both of interference between the plates and of separated vortices, was developed to give adequate accuracy in analyzing beating wings in near fling. Introduction The fling mechanism was discovered by Weis-Fogh (973) 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 (973) analyzed this mechanism theoretically by using inviscid theory. He ignored the vortices shed at the outer edges. Maxworthy (979) 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 (982) and Wu and Hu-chen (984) introduced a concentrated vortex shed at the outer edge into an inviscid flow analysis and indicated the importance of this vortex. Haussling (979) solved the two-dimensional Navier Stokes equations at Re=3 and obtained the instantaneous streamlines and vorticity lines. Key words: near fling, hydrodynamics, vortex method, hydrodynamic interference, quasi-steady analysis, flight.
28 S. SUNADA AND OTHERS Spedding and Maxworthy (986) measured the instantaneous lift forces on a pair of wings in fling. Ellington (984a) 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, 973; Ellington, 984b) 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, 99) 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. Abbreviations [A n,m ] matrices of influence coefficients caused by bound vortices on the plate (m ) [B n,m ] matrices of influence coefficients caused by bound vortices on the other plate (m ) 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 ) c v chord length of plate for two-dimensional visualization test (m) d non-dimensional distance between mirror and rotation axis, d /x tip d distance between mirror and rotation axis (m) dm f added mass of two-dimensional plate (kg) [D n, ] 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) f f 6 shape factors (m 4 or m 5 ) g acceleration due to gravity (9.8 ms 2 ) I moment of inertia around rotation axis (kg m 2 ) K. k F,., k M, non-dimensional parameter, b(y)/x tip quasi-steady force coefficients at.. = L length between centre axis of load cell and rotation axis (m) l, l 2 positions of weight (m) height of weight (m) l 3
Analysis of near fling 29 l w M f M Y m m p m w p R Re t t p t w u S, v S distance between rotation axis and gravitational centre of weight (m) hydrodynamic moment around rotation axis (Nm) moment around Y-axis measured by a load cell (Nm) vortex number mass of plate (kg) mass of weight (kg) angle indicated in Figs 2 and 3 (rad) non-dimensional parameter, c tip /x tip Reynolds number time (s) thickness of plate (m) thickness of weight (m) x and y components of inflow velocity to bound vortex which is not parallel with x-axis (ms ) V F shape factor which is proportional to added mass (m 4 ) V M shape factor which is proportional to added moment of inertia (m 5 ) V p volume of plate (m 3 ) V w volume of weight (m 3 ) v T y component of inflow velocity into bound vortex which is parallel with x-axis (ms ) 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) opening angle as shown in Fig. 3 (rad) initial opening angle (rad) circulation of bound vortex (m 2 s ) l S length of vortex filament which is not parallel to x-axis (m) l T length of vortex filament which is parallel to x-axis (m) S panel area (m 2 ) t time step (s) difference of the non-circulatory component of velocity potential between the upper and lower surface (m 2 s ) kinematic viscosity (m 2 s ) f density of fluid (kgm 3 ) p density of plate (kgm 3 ) w density of weight (kgm 3 ) velocity potential (m 2 s ) inclination angle of bound vortex as shown in Fig. 7A (rad) angular velocity in two-dimensional flow visualization test (rad s )
22 S. SUNADA AND OTHERS 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 normal force or moment due to dynamic pressure 2 normal force or moment due to impulsive pressure Apparatus and method used to measure force in near fling The experimental apparatus is shown in Fig.. An acrylic plate was suspended from a load cell in front of a mirror in a water tank (6mm 6mm 8mm). 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. The geometrical relationship between the mirror and the plate is shown in Fig. 3. The nondimensional distance between the rotation axis of the plate and the mirror, d=d /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 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 was recorded by a video camera. The angular velocity. and angular acceleration.. were obtained by differentiating and., respectively. The forces in two directions, F X and F Z, and moment, M Y =F X L, 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. The measured forces, F X and F Z, include the inertial forces due to the plate motion. Therefore, the normal and tangential forces due to hydrodynamic force, F N and F S, are estimated from F X and F Z as follows:
Analysis of near fling 22 Load cell Plate Mirror Water tank Movie camera Still camera Fig.. Experimental apparatus and its arrangement. Still camera 2 Plate element 2 b(y) Plate A Plate B Rotation axis Rotation axis p l l2 l w l Plate element 2 xcg tw l 3 xtip Weight b(y) t w l2 x CG xtip p lw l 3 Weight x c(x) c tip t p c(x) Plate element x Plate element ctip t p Fig. 2. Test plates. The gravitational forces acting on the plate m p g and the weight m w g were cancelled by the adjustment on zero point of the load cell F Z before the measurement. The plate rotated freely around the rotation axis. The equation of rotational motion, therefore, is given by: (I p + I w ).. ( p f )V p x CG gsin ( w f )V w l w gsin( + p)+ M f =. (2)
222 S. SUNADA AND OTHERS Table. Geometrical characteristics of test plates and of the weight Variable Symbol Units Plate A Plate B Test plate Mass m p kg 2.98 2 2.5 2 Length x tip m.2. Chord length at tip c tip m..9 Plate thickness t p m 2. 3 2. 3 Centre of gravity x CG m 5.6 2 6.66 2 Area S p m 2.3 2 9.55 3 Volume V p m 3 2.38 5 2. 5 Density p kg m 3.25 3.25 3 Moment of inertia I p kg m 2.2 4.2 4 f m 4 5. 5 7.2 5 f 2 m 5 3.8 6 5.7 6 Shape factor f 3 m 4 2.4 5 2.4 5 f 4 m 5 2. 6.9 6 f 5 m 4 5.6 5 3. 5 f 6 m 5 3.5 6 2. 6 Variable Symbol Units Values Weight Mass m w kg 9.4 3 Position l m 2.5 2 Position l 2 m 3.7 2 Height l 3 m 2. 2 l w =/2 (l +l 2 ) 2 +l 2 3 m 3.3 2 p=tan [l 3 /(l +l 2 )] rad.3 Thickness t w m 3.4 3 Volume V w m 3 8.2 7 Density w kg m 3.5 4 Moment of inertia I w kg m 2. 5 Non-dimensional distance, d Table 2. Test cases Initial opening angle,.2.6.9, /2.9 The hydrodynamic moment around the rotation axis is derived from the above equation as:
Analysis of near fling 223 Load cell MY d L FN + mpxcg mpxcg 2 p F S mwlw FX m w l w 2 Mf Mirror FZ Fig. 3. Geometrical arrangements of the plate and the mirror and the definitions of forces and moments. The two plates A and B were designed to have almost the same values in these terms at every opening angle. The common relationship between and M f was obtained, therefore, for both plates A and B and is shown in Fig. B and Fig. B. Flow visualization Edwards and Cheng (982) and Wu and Hu-chen (984) 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, 984a; Spedding and Maxworthy, 9 8 6 ). 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=.2 and d=, were set
224 S. SUNADA AND OTHERS for each plate shape. The experimental apparatus is shown in Fig.. The pictures of flow were taken by two still cameras, cameras and 2. The wake was made visible by use of the dye Methylene Blue (Sunada et al. 989). Fig. 4A,B shows photographs for the cases of distance d=.2 and d=, respectively, for plate A with the initial opening angle =. The photographs taken by camera and by camera 2 show the flow pattern at 2 and.7, respectively. There is little difference of vortex systems between the two distances, d=.2 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=.2 and d=, respectively, for plate B. There is little difference in the vortex systems at the two distances, d=.2 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 6mm, rotates around an axis which is perpendicular to the water surface. The non-dimensional distance between the rotation axis and the wall is d=d /c v =.4. Pictures of the flow pattern were taken by a still camera set above the water tank. The rotation speed was about /2rad s and the Reynolds number, Re=.75c v 2 /, was 3 3. 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, 986). The behaviour of the vortices is different, however, from that in the sketch of two-dimensional near fling by Ellington (Fig. 4c of Ellington, 984a), where the inner vortices were left in the water and the outer vortices were located near the plate surface. Numerical calculation 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. It is assumed that the pressure fields around the plates can be predicted by the use of the
Analysis of near fling 225 A B Vortex Vortex Plate Plate Rotation axis Rotation axis Fig. 5. Sketch of flow pattern of three-dimensional plates. (A) Flow pattern for plate A; (B) flow pattern for plate B. potential flow method. The velocity potential of the flow field around a pair of plates satisfies the following Laplace equation: 2 =. (4) As in Theodorsen s method (Theodorsen, 934), this velocity potential is divided into two components, a non-circulatory component NC and a circulatory component C as follows: = NC + C. (5) The non-circulatory component satisfies the boundary condition on the surfaces of the plates by using bound vortices and without vortices shed into the flow. The sum of the circulation of these bound vortices is zero. Therefore, this component of the velocity potential was expressed by using sinks and sources instead of bound vortices in Theodorsen s method, and it was called a non-circulatory component. The circulatory component is generated by the vortices shed into the flow and does not change the boundary condition on the surfaces of the plates. The non-circulatory component satisfies the Laplace equation, that is: 2 NC =. (6) The solution of the above equation is obtained by using a vortex lattice method (Levin, 984; Katz, 984). A plate is composed of small panels as shown in Fig. 7A. The orders of panels in the x- and y-directions are represented by j and k, respectively, as shown in Fig. 7B. The mth panel is defined as: m = k max (j )+ k. (7) This panel has the mth non-circulatory component of bound vortex, NC,m,ic and the mt h collocation point is placed at the centre of the panel. The direction of circulation of the
226 S. SUNADA AND OTHERS
Analysis of near fling 227 bound vortex is that of the right-handed revolution along the arrow shown in Fig. 7B. The suffix i c represents the current time step. The following equation is derived with the use of the boundary condition that the flow moves along the plate surface at each collocation point: The first term in equation 8 represents the induced velocity at a collocation point generated by the bound vortices of a pair of plates. The influence coefficient matrices [A n, m ] and [B n, m ] represent the induced velocity on the nth collocation point generated by the unit non-circulatory component of bound vortices of the mth panels on the same plate and the other plate, respectively. Hence, [B n, m ] depends on the opening angle and the distance d. When an isolated plate is considered, [B n, m ] becomes the zero matrix. The second term represents the moving velocity of each collocation point, and the matrix [D n, ] is defined as the distance between the collocation point and the rotation axis. By solving equation 8 at each time step, the non-circulatory component of the bound vortex NC,m,ic is obtained. The total velocity potential, which is the sum of the circulatory component and the noncirculatory component, is obtained by solving equation 4. Like the model for the noncirculatory component, the mth panel has the mth bound vortex m,ic as shown in Fig.7C. As confirmed by the results of the flow visualization test, no bound vortex is settled on the edge BC, and this means that the Kutta condition is satisfied at this edge. The vortices are also generated from the inner edges, AB and AC, according to the flow visualization tests. The vortices from the inner edges, which are near the plate surface, induce less velocity along the plate surface than that induced by the vortices from the outer edges, however, so the effect of the inner vortices on the pressure distribution of the plate is less than that of the outer vortices. The vortices are, then, represented by vortices generated from the outer edge BC alone in the present numerical calculation. The wake, which is composed of these vortices generated from the outer edge, is placed on the surface of the cylinder swept by the edge BC, as shown by Fig. 7D. This surface of the cylinder, which is shaded in Fig. 7D, is extended to a plate, as shown in Fig. 7E. The wake is represented by the sum of the rectangular vortex elements as shown in Fig. 7E. Each vortex element has a continuously distributed vortex sheet in the element. The strength of this vortex sheet is defined by the circulation at the four corners of the element, kmax(jmax )+k,i, kmax(jmax )+k,i, kmax(jmax )+k,i, kmax(jmax )+k,i, as shown in F i g. 7E. The method used to calculate the velocity induced by a rectangular vortex sheet in the wake element is given by Johnson (98). The following equation is derived from the boundary condition that the flow moves along the plate surface at each collocation point:
228 S. SUNADA AND OTHERS A A y S j,k l S,j,k k Collocation points + Points where force is calculated lt,j,k j B C j,k x B A C A NC(j=,k=k) Collocation point (j=,k=k) Collocation point NC,m,ic = NC(j=j,k=k) C m,i c = (j=j,k=k) C NC(j=j max,k=k) (j=jmax,k=k) NC(j=j max,k=k) (j=jmax,k=k) B C B C Vortex sheet D Vortex sheet Plate at t= C C B A A B Plate at t=t Plate at t=t Rotation axis Fig. 7
Analysis of near fling 229 E A Rotation axis Plate i=ic i=i c B C kmax(jmax )+k,i k max (j max )+k,i k max (j max )+k,i k max (j max )+k,i i= i= i= i= i=2 Vortex sheet i=i c i=i c B k= k= Plate C k=kmax+ k=kmax A Rotation axis Fig. 7. Model for numerical calculation. (A) Panel shapes on a plate and arrangement of collocation points and points where force is calculated; (B) arrangement of non-circulatory bound vortex NC; (C) arrangement of total bound vortex ; (D) wake geometry assumed in numerical calculation; (E) circulation of separated vortices. Again, the first term represents the induced velocity at the collocation point generated by the bound vortices of a pair of plates. The second term represents the moving velocity of each collocation point. The third term represents the induced velocity of the collocation point generated by the vortices shed into the flow. By solving equation 9 at each time step, the circulation of the bound vortex m,ic is obtained. Because the non-circulatory component of the mth bound vortex was obtained using equation 8, the circulatory component of mth bound vortex is then given by: C,m,ic = m,ic NC,m,ic. ( )
23 S. SUNADA AND OTHERS The total normal force F N,ic and total moment around rotation axis M f,ic are given by: where The first terms, F N,ic and M f,ic, and the second terms, F N2,ic and M f2,ic, in equation are due to dynamic pressure and impulsive pressure of the unsteady Bernoulli s equation, respectively (Lamb, 932). The quantities p and q in the suffix (p, q) of F N(p,q),ic and M f(p,q),ic in equation 2 are taken to be either a non-circulatory component NC or a circulatory component C. The quantity p indicates the type of vortex component that generates the induced velocity at the point of the bound vortex. The quantity q indicates the type of component of the bound vortex. When an isolated plate is considered, the induced velocity at the point of the bound vortex that is generated by the non-circulatory component is always perpendicular to the plate surface. This is because the non-
Analysis of near fling 23 circulatory component of the vortex is distributed only on the plate surface. The following components of normal force and moment for an isolated plate are therefore given by: Calculation of added mass and added moment of inertia The normal force and the moment around rotation axis at t are mainly due to impulsive pressure of the non-circulatory component. The difference in the noncirculatory component of velocity potential between the upper and lower surfaces NC is proportional to the angular velocity.. Therefore, NC is expressed as: NC =. G( ). (4) When an isolated plate is considered, G is a constant instead of a function of opening angle. The pressure difference distributed on the plate due to the non-circulatory component is given by: The non-circulatory components of the normal force and the moment around the rotation axis acting on the plate are obtained by integrating the pressure difference on the plate surface as: where As shown in the above equations, the non-circulatory component of the normal force (the moment) is composed of two components. One is proportional to the angular acceleration... The quantities V F and V M are shape factors; they are proportional to added mass and added moment of inertia, respectively. The second component is proportional to.2. When both the distance d and the opening angle are small, V F / and V M / are large, with the result that each of the second components is not small. This will be shown quantitatively later. The non-circulatory components of the normal force and the moment are proportional to angular acceleration.. only at the very beginning of the motion t. This is because the angular velocity. is nearly rad s at this moment, and the second
232 S. SUNADA AND OTHERS components of the force and the moment are negligible. The shape factors V F and V M can, therefore, be obtained from values of F N (t=), M f (t=) and.. (t=) by using the following equations: Normal force F N (t=) and moment M f (t=) acting on a plate that begins to rotate at angular acceleration.. (t=) are obtained by using the present numerical calculation method and from experimental data. The shape factors can then be given by both the numerical and experimental methods. Results Fig.8A D shows the motion of plates that occurred when the initial opening angle was. Time histories of opening angle for plates A and B are shown in Fig. 8A and B, respectively. Fig. 8C and D indicate, respectively, how. and.. varied with for two 4 2 A d=.2 d= d=.6 d=.9 d=.9.5 C Plate B, d= Plate A, d=.2.25.5.75.25.5.75 4 B D Plate B, d= Plate A, d=.2 2 d=.2 d= d=.6 d=.9 d=.9.25.5.75.25.5.75 Opening angle, (rad) Fig. 8. Time histories of plate motion ( =). (A) Opening angle of plate A; (B) opening angle of plate B; (C) angular velocity. ; (D) angular acceleration.
Analysis of near fling 233 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.. =rad s 2. Fig. 9A C shows the motion of plates with = /2 for two specific cases. It is observed that there is little difference among all the cases at t=. The measured and calculated forces and moments acting on a plate are indicated in Fig. A D for = and in Fig. A,B for = /2. The shaded areas in Fig. A and Fig. A show the normal force F N and the tangential force F S, which were calculated with equation 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. B and Fig. B show the experimental results of the relationship between and M f, which is the same for both plate A and plate B, as stated before. The solid lines in Fig. A,B and Fig. A,B show the analytical results for test cases with d=.2, and the broken lines in Fig. A,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 N and M f, 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. A,B and Fig. A,B show the results obtained by the numerical calculation, where the measured plate movement for d=.2 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 is near /2. Fig.C,D shows the circulation of the bound vortex which is generated along y= c tip /4 at =.3, and each component of the normal force indicated in equation 2. 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=.2. The narrow lines express the results obtained by using the calculation model, where the measured plate movement for d=.2 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 C near the rotation axis. Because the timewise change of integrated 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 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
234 S. SUNADA AND OTHERS 2 A Plate A, d=.2 Plate B, d=.2 Plate A, d= Plate B, d=.5.75 B Plate A, d=.2.5 Plate B, d=.5.75 4 C 2 Plate B, d= Plate A, d=.2 2.5.75 Opening angle, (rad) Fig. 9
Analysis of near fling 235 Fig. 9. Time histories of plate motion ( = /2). (A) Opening angle ; (B) angular velocity. ; (C) angular acceleration. velocity component along the surface of the plate is made larger by the interference between plates of a pair. The bound vortex circulation near the rotation axis on the plate is also increased by the interference. Hence, the value of F N(C, C)+F N(C, N C) is increased, because the value of F N(C, C)+F N(C, N C) is proportional to the multiple of the bound vortex circulation and the induced velocity component along the plate surface. The other component of normal force, F N(N C, N C), the value of which is zero for an isolated plate as indicated in equation 3, has a positive value. Therefore, the interference increases the normal force due to dynamic pressure, F N(C, C)+F N(C, N C)+F N(N C, N C), and the moment around the rotation axis, M f(c, C)+M f(c, N C)+M f(n C, N C). This is indicated as the difference at =.3 between the solid line and the chain line in Fig. A,B. When the opening angle increases, this interference between the two plates of a pair becomes small. It is also observed in Fig. D that the value of F N(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. 2A E. Fig. 2A,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, = and = /2, are shown. When the opening angle 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 i n F i g. 2B 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 = /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. 2C and D, respectively. The results indicate good agreement between measurement and calculation. Fig. 2E shows the variation of added mass with the opening angle 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.2 ( V F / ) and.2 ( V M / ) in equation 6. The normal force due to the impulsive pressure of the non-circulatory component and each of its components given by equation 6 are shown in Fig. 3 with d=.2. The chain line indicates the c a l c u l a t e d F N2(N C), where the measured plate movement for d=.2 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 < <. only. The component.2 ( V F / ) reduces the total value of the normal force due to the impulsive pressure of the non-
236 S. SUNADA AND OTHERS A Experiment, F N.5 FN2 FN Experiment, FS.5.25.5.75 Opening angle, (rad) B Experiment 5 M f2 Mf.25.5.75 Opening angle, (rad) C C NC.25.5.75. Position along y= ctip/4, 2x/xtip Fig.
Analysis of near fling 237 D F N(C,NC)+F N(C,C) 5 F N2(NC) F N2(C) FN(C,C) F N(NC,NC) F N(C,NC) 2.5.25.5.75 Opening angle, (rad) Fig.. Time histories of normal and tangential forces and moment around rotation axis ( =). (A) Normal force, FN and tangential force, FS. (B) Moment around rotation axis, Mf. Hatched area or, experimental results;, calculation result with d=.2;, calculation result with d= ; -, calculation result where the effect of the other (e.g. the image) plate is ignored with d=.2. (C) Circulation distribution of bound vortex on y= ctip/4 along the x-axis at =.3rad. (D) Component of normal force. Bold lines, calculation result with d=.2; narrow lines, calculation result where the effect of the other plate is ignored with d=.2. circulatory component, and this total value with the interference is smaller than that without the interference during the period >.. In summary, the beating motion, where both the distance d and the opening angle are small, makes it possible to increase the added mass of a wing. The normal force due to the impulsive pressure of the noncirculatory component, however, is increased during the very limited period in which the opening angle is very small. Discussion Capability of simple analysis A simple model was proposed for analyzing the wing-beating flight of insects (Weis- Fogh, 973; Ellington, 984b). 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 ) 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= and.. =rad s 2, are selected. The force and moment acting
238 S. SUNADA AND OTHERS A Experiment, F N.5 F N2 FN Experiment, F S.5.5.75 B 5 Experiment M f2 Mf.5.75 Opening angle, (rad) Fig.. Time histories of normal and tangential forces and moment around rotation axis ( = /2). (A) Normal force, FN and tangential force, FS. (B) Moment around rotation axis, Mf. Hatched area or, experimental results;, calculation result with d=.2; -, calculation result where the effect of the other (e.g. the image) plate is ignored with d=.2. on a plate are caused by the added mass alone at t=, and by the quasi-steady force alone at.. =rad s 2. 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. Calculation at t= It is assumed that a plate element has an added mass equal to the two-dimensional plate
Analysis of near fling 239 of the same width c(x). A plate element, then, has the following added mass (Lamb, 932): dm f = f {c(x)/2} 2 dx. (9) The normal force F N (t=) and the moment around the rotation axis M f (t=), which are caused by the angular acceleration of the plate at the beginning of motion, are obtained by integrating their components acting on the plate element as follows: The shape factors, f and f 2, are defined as: The non-dimensional shape factors are given by: where R = c tip /x tip. (25) As in the previous simple model of Weis-Fogh or Ellington, plate element 2 is assumed to be part of the two-dimensional plate of the same width b(y) rotating with angular acceleration... The normal force and the moment around the rotation axis generated on this element are given by (Lamb, 932): where
24 S. SUNADA AND OTHERS 5 A Plate A Plate B Plate A Plate B = = /2..2 B Plate A Plate B Plate A Plate B = = /2..2 2 C Plate A Plate B Plate A Plate B = = /2..2 Non-dimensional distance, d Fig. 2
Analysis of near fling 24 2 D Plate A Plate B Plate A Plate B = = /2..2 Non-dimensional distance, d 5 E Plate A Plate B Plate A Plate B d=.2 d=.9.5..5 Opening angle, (rad) Fig. 2. Added mass of plate A and plate B. (A) Shape factor VF versus non-dimensional distance d; (B) centre of pressure VM/VFxtip versus d; (C) ratio between measured and calculated shape factor VF; (D) ratio between measured and calculated shape factor VM; (E) shape factor VF versus opening angle. The normal force and moment around the rotation axis at t= are given by integrating these compoments of plate element 2 as follows:
242 S. SUNADA AND OTHERS f V F F N2(NC) f V F, without the effect of the other plate F N2(NC)= f { V F + 2( V F / )} f 2( V F / ).5 Opening angle, (rad) Fig. 3. Normal force due to impulsive pressure of non-circulatory component., normal force with d=.2, FN2(NC)= f{ VF+.2 ( VF/ )};, normal force component with d=.2, f VF; --, normal force component with d=.2, f.2 ( VF/ ); -, normal force where the induced velocity generated by the other (e.g. the image) plate is ignored with d=.2, FN2(NC) f VF. where The non-dimensional shape factors are given by: Comparing equations 2 and 28 with equation 8, it should be noted that shape factor f or f 5 corresponds to shape factor V F and that shape factor f 2 or f 6 corresponds to shape factor V M. Shape factors V F and V M are dependent not only on plate shape but also on opening angle and distance d, because the interference effect between a pair of plates can be considered for calculating these factors by using the vortex lattice method. In
Analysis of near fling 243 4 A 2 C 2 f /x tip 4 f 5 /x tip 4 V F /x tip 4 f /x tip 4 f 5 /x tip 4 V F /x 4 Plate A tip 2 3 Plate B 2 3 2 B f 2 /x tip 5 f 6 /x tip 5 D V M /x tip 5 f2/xtip 5 f6/xtip 5 V M /x 5 Plate A tip 2 3 Plate B 2 3 R=c tip /x tip Fig. 4. Non-dimensional shape factors. (A) Non-dimensional shape factors of rectangular plate, f /x tip 4, f 5 /x tip 4 and V F /x tip 4 ; (B) non-dimensional shape factors of rectangular plate, f 2 /x tip 5, f 6 /x tip 5 and V M /x tip 5 ; (C) non-dimensional shape factors of triangular plate, f /x tip 4, f 5 /x tip 4 and V F /x tip 4 ; (D) non-dimensional shape factors of triangular plate, f 2 /x tip 5, f 6 /x tip 5 and V M /x tip 5. contrast, shape factors calculated by the simple methods, f, f 2, f 5 and f 6, are dependent on plate shape alone. F i g. 4A D shows the variations of the non-dimensional shape factors with the variation of the non-dimensional plate geometry ratio R=c t i p /x t i p. The results for the two simple models, obtained by use of equations 23, 24, 3 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, 973; Ellington, 984b) is in good agreement with the vortex lattice method when the ratio R is less than.5. In contrast, the present simple model approach provides acceptable solutions when the ratio R i s large (R>2). This means that the flow around the plate at the beginning of the plate motion is parallel to plate element for small values of R and to plate element 2 for large values of R near the beginning of the plate motion, t. It is also observed that neither of the simple model approaches provides acceptable predictions for an intermediate value of the ratio R ( ), 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
244 S. SUNADA AND OTHERS precisely a low aspect ratio wing such as a butterfly wing or the interference between two beating wings. Calculation at = ( quasi-steady hydrodynamic force) The previous simple model (Weis-Fogh, 973) estimates the quasi-steady hydrodynamic force at the instant.. =rads 2 as follows. It is assumed that the velocity induced by the vortices in the wake is ignored when counting the inflow velocity towards plate element ; that is, the relative velocity of the element is defined by the angular velocity. alone. In addition, the quasi-steady force coefficient of the element is assumed to be k F,. for calculation of the normal force or k M,. for calculation of the moment around the rotation axis, and both the coefficients are assumed to be constant for all elements. The normal force F N (.. =) and the moment around the rotation axis M f (.. =) are expressed by integrating these components acting on the plate element as follows: where The values of k F,. and k M,. are defined by substituting the measured values of force and moment around the rotation axis for F N and M f in equations 33 and 34. The results are shown in Table 3. It is indicated that these values are independent of the distance d. This means that the interference between two plates of a pair has a negligible effect on the values of k F,. and k M,. at.. =rad s 2, that is, /2. Differences in the values of k F,. or k M,. are observed in those cases where is different, even though the Reynolds number Table 3. Values of k F, and k M, at = Plate ( =) Re k F, k M, d A.5 5.5 3 4.6 4. A /2 2. 5.5 3 3.2 3.2.2,.6,.9 B.5 7 3 4.6 4.2.9, B /2 2. 7 3 3.2 3.4
Analysis of near fling 245 is the same. These differences are caused by the induced velocity generated by the vortices in the wake. In addition, these values of k F,. and k M,. are much greater than the steady drag coefficient,.98, acting on a two-dimensional plate in the normal flow at a Reynolds number of 4 (Hoerner, 958). This inconsistency observed in the calculated values of k F,. and k M,. shows the limitation of the applicability of the previous simple model (Weis-Fogh, 973) and indicates that the flow around the rotating plate is very different from the assumed two-dimensional flow. However, this simple method is useful in providing rough estimates of the hydrodynamic force generated by insect wings, if the modified hydrodynamic force coefficient (e.g. 4) is adopted. Conclusion 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 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 ) or perpendicular to the rotation axis (plate element 2). The simple method using plate element can estimate the added mass of isolated beating wings when the nondimensional plate geometry ratio R is less than.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
246 S. SUNADA AND OTHERS wings when the geometry ratio is greater than 2. The geometry ratio of butterfly wings is about, 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 (Weis-Fogh, 973; Ellington, 984b) 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, 99). 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. Appendix 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. Load cell 2F Z Plate Plate 2d Fig. 5. Experimental arrangement for method I.
Analysis of near fling 247 Method I Method II.5 5 Time, t (s) Fig. 6. Comparison of vertical forces measured using the two different methods. Fig. 5 shows the arrangement of the experimental apparatus for method I. The force in direction Z, 2F Z in equation, 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=.2, is selected for this experiment. The solid and broken lines in Fig. 6 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. References DUDLEY, R. (99). Biomechanics of flight in neotropical butterflies: aerodynamics and mechanical power requirements. J. exp. Biol. 59, 335 357. EDWARDS, R. H. AND CHENG, H. K. (982). The separation vortex in the Weis-Fogh circulationgeneration mechanism. J. Fluid Mech. 2, 463 473. ELLINGTON, C. P. (984a). The aerodynamics of flapping animal flight. Am. Zool. 24, 95 5. ELLINGTON, C. P. (984b). The aerodynamics of hovering insect flight. II. Morphological parameters. Phil. Trans. R. Soc. Lond. B 35, 7 4. HAUSSLING, H. J. (979). Boundary-fitted coordinates for accurate numerical solution of multibody flow problems. J. comp. Physiol. 3, 7 24. HOERNER, S. F. (958). Fluid-Dynamic Drag. Midland Park: Published by the author. JOHNSON, W. (98). Helicopter Theory. Princeton: Princeton University Press. KATZ, J. (984). Lateral aerodynamics of delta wings with leading-edge separation. AIAA Journal 22(3), 323 328. LAMB, H. (932). Hydrodynamics. Cambridge: Cambridge University Press. LEVIN, D. (984). A vortex-lattice method for calculating longitudinal dynamic stability derivatives of oscillating delta wings. AIAA Journal 22(), 6 2. LIGHTHILL, M. J. (973). On the Weis-Fogh mechanism of lift generation. J. Fluid Mech. 6, 7.
248 S. SUNADA AND OTHERS MAXWORTHY, T. (979). Experiments on the Weis-Fogh mechanism of lift generation by insects in hovering flight. I. Dynamics of the fling. J. Fluid Mech. 93, 47 63. SPEDDING, G. R. AND MAXWORTHY, T. (986). The generation of circulation and lift in a rigid twodimensional fling. J. Fluid Mech. 65, 247 272. SUNADA, S., KAWACHI, K. AND AZUMA, A. (989). Vortices generated by butterfly s wings at take-off phase. In Flow Visualization, vol. V, pp. 2 7. Washington: Hemisphere. THEODORSEN, T. (934). General theory of aerodynamic instability and the mechanism of flutter. NACA Rep. 496. WEIS-FOGH, T. (973). Quick estimates of flight fitness in hovering animals, including novel mechanisms for lift production. J. exp. Biol. 59, 69 23. WU, J. C. AND HU-CHEN, H. (984). Unsteady aerodynamics of articulate lifting bodies. AIAA Paper no. 284.