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 18 June 1993
SummaryA 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 rotationaxes of the two plates of each pair was varied. The motion of the plates as well as theforces and the moment were measured, and the interference between the two plates of apair was studied. In addition, a method of numerical calculation was developed to aid inthe understanding of the experimental results.
The interference between the two plates of a pair, which acted to increase both theadded mass of each plate and the hydrodynamic force due to dynamic pressure, was notedonly when the opening angle between the plates was small. The hydrodynamic forceswere strongly influenced by separated vortices that occurred during the rotation. Amethod of numerical calculation, which took into account the effect both of interferencebetween the plates and of separated vortices, was developed to give adequate accuracy inanalyzing beating wings in ‘near fling’.
Introduction
The ‘fling’ mechanism was discovered by Weis-Fogh (1973) through observing theflight of the wasp Encarsia formosa. The mechanism was paid increasing attentionbecause a much larger force than the quasi-steady-state value was obtained byobservation of the flight of insects that used these mechanisms. Lighthill (1973) analyzedthis mechanism theoretically by using inviscid theory. He ignored the vortices shed at theouter edges. Maxworthy (1979) performed a series of experiments and pointed out thatthe magnitude of circulation generated during ‘fling’ is much larger than that calculatedby Lighthill. Edwards and Cheng (1982) and Wu and Hu-chen (1984) introduced aconcentrated vortex shed at the outer edge into an inviscid flow analysis and indicated theimportance of this vortex. Haussling (1979) solved the two-dimensional Navier–Stokesequations at Re=30 and obtained the instantaneous streamlines and vorticity lines.
Spedding and Maxworthy (1986) measured the instantaneous lift forces on a pair ofwings in ‘fling’. Ellington (1984a) pointed out that ‘near fling’ motion is also observed inthe flight of some insects. In these studies, two-dimensional ‘fling’ and ‘near fling’ wereassumed, but three-dimensional effects were not considered except by Maxworthy.Quantitative study of three-dimensional motion is important in analyzing the actual flightof 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 orderto 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. Thehydrodynamic characteristic of ‘fling’ can be extrapolated from the experimental resultsof ‘near fling’. A comparison of the present three-dimensional analysis with the previoustwo-dimensional analysis for beating wings (Weis-Fogh, 1973; Ellington, 1984b) ispresented. The results of the comparison will make clear the limitation of the applicabilityand the accuracy of the two-dimensional analysis that has been used to analyze beatingwings. The large quasi-steady force coefficient obtained by two-dimensional analysis(Dudley, 1991) is verified by the present research. The numerical calculation techniquedeveloped in this paper makes it possible to analyze real insect flight where ‘fling’ or‘near fling’ is used.
Abbreviations
[An,m] matrices of influence coefficients caused by bound vortices on the plate(m−1)
[Bn,m] matrices of influence coefficients caused by bound vortices on the otherplate (m−1)
b, c chord length of plate for three-dimensional test as shown in Fig. 2 (m)[Cn,4] matrix of influence coefficients by vortex elements in wake (m−1)cv chord length of plate for two-dimensional visualization test (m)d non-dimensional distance between mirror and rotation axis, d0/xtip
d0 distance between mirror and rotation axis (m)dmf added mass of two-dimensional plate (kg)[Dn,1] matrix of collocation points (m)FN normal component of hydrodynamic force acting on plate (N)FS tangential component of hydrodynamic force acting on plate (N)FX, FZ X and Z components of force measured by a load cell (N)f1–f6 shape factors (m4 or m5)g acceleration due to gravity (9.81 ms−2)I moment of inertia around rotation axis (kg m2)K non-dimensional parameter, b(y)/xtip
kF,a., kM,a
. quasi-steady force coefficients at a..=0
L length between centre axis of load cell and rotation axis (m)l1, l2 positions of weight (m)l3 height of weight (m)
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lw distance between rotation axis and gravitational centre of weight (m)Mf hydrodynamic moment around rotation axis (Nm)MY moment around Y-axis measured by a load cell (Nm)m vortex numbermp mass of plate (kg)mw mass of weight (kg)p angle indicated in Figs 2 and 3 (rad)R non-dimensional parameter, ctip/xtip
Re Reynolds numbert time (s)tp thickness of plate (m)tw thickness of weight (m)uS, vS x and y components of inflow velocity to bound vortex which is not parallel
with x-axis (ms−1)VF shape factor which is proportional to added mass (m4)VM shape factor which is proportional to added moment of inertia (m5)Vp volume of plate (m3)Vw volume of weight (m3)vT y component of inflow velocity into bound vortex which is parallel with
x-axis (ms−1)X, Y, Z earth-fixed coordinate systemx, y, z plate-fixed coordinate systemxC x-coordinate at collocation points (m)xCG x-coordinate of gravitational centre of plate (m)xS x-coordinate of centre of bound vortex filament which is not parallel with
x-axis (m)xT x-coordinate of centre of bound vortex filament which is parallel with x-axis
(m)a opening angle as shown in Fig. 3 (rad)a0 initial opening angle (rad)G circulation of bound vortex (m2 s−1)DlS length of vortex filament which is not parallel to x-axis (m)DlT length of vortex filament which is parallel to x-axis (m)DS panel area (m2)Dt time step (s)Df difference of the non-circulatory component of velocity potential between
the upper and lower surface (m2 s−1)n kinematic viscosity (m2 s−1)rf density of fluid (kgm−3)rp density of plate (kgm−3)rw density of weight (kgm−3)f velocity potential (m2 s−1)w inclination angle of bound vortex as shown in Fig. 7A (rad)V angular velocity in two-dimensional flow visualization test (rad s−1)
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Subscripts or superscripts
C circulatory componentf value of fluidi i-th time stepic current time stepj vortex number in j-direction as shown in Fig. 7Ak vortex number in k-direction as shown in Fig. 7Amax maximum valueNC non-circulatory componentp value of platetip value at tipw value of weight1 normal force or moment due to dynamic pressure2 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. 1. An acrylic plate was suspended from aload cell in front of a mirror in a water tank (600mm×600mm×800mm). The mirrorreflected an image of the plate. This made it possible for the flow around the plate in frontof the mirror to simulate the flow around the two plates of a pair. A verification of thisexperimental approach, a comparison with an experiment using a real pair of rotatingplates, is given in the Appendix. Two different shapes of plates, rectangular (plate A) andtriangular (plate B), were used. A small weight was attached to each plate in order tocreate an initial moment. The configuration of the plate and the weight are shown inFig. 2, and the dimensions of the plates and the weight are given in Table 1. Thegeometrical relationship between the mirror and the plate is shown in Fig. 3. The non-dimensional distance between the rotation axis of the plate and the mirror, d=d0/xtip, wasvaried in order to investigate the interference between the two wings of a pair in ‘nearfling’ motion. In addition, the effect of varying the initial opening angle a0 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 thenremoved and the plate began to rotate around the axis by gravitational force. The resultantvariation of the opening angle a was recorded by a video camera. The angular velocity a
.
and angular acceleration a..
were obtained by differentiating a and a., respectively. The
forces in two directions, FX and FZ, and moment, MY=FXL, were measured by the loadcell as shown in Fig. 3. The value MY/L was compared with the directly measured forceFX to confirm the accuracy of the values of FX.
The measured forces, FX and FZ, include the inertial forces due to the plate motion.Therefore, the normal and tangential forces due to hydrodynamic force, FN and FS, areestimated from FX and FZ as follows:
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The gravitational forces acting on the plate mpg and the weight mwg were cancelled by theadjustment on zero point of the load cell FZ before the measurement. The plate rotatedfreely around the rotation axis. The equation of rotational motion, therefore, is given by:
Fig. 1. Experimental apparatus and its arrangement.
Water tankMirror
Plate
Still camera 1
Movie camera
Still camera 2
Load cell
xx
b(y) b(y)
c(x)
ctipctiptp
tp
xtip
xtiptw
tw
l1 l1l2 l2
l3
p p
Plate A Plate B
lw lw
l3
Plate element 1 Plate element 1
Weight Weight
Rotation axis Rotation axisPlate element 2
Plate element 2xCG xCG
c(x)
Fig. 2. Test plates.
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The hydrodynamic moment around the rotation axis is derived from the above equationas:
222 S. SUNADA AND OTHERS
Table 1. Geometrical characteristics of test plates and of the weight
Variable Symbol Units Plate A Plate B
Test plateMass mp kg 2.98×10−2 2.50×10−2
Length xtip m 1.12×10−1 1.00×10−1
Chord length at tip ctip m 1.01×10−1 1.91×10−1
Plate thickness tp m 2.1×10−3 2.1×10−3
Centre of gravity xCG m 5.6×10−2 6.66×10−2
Area Sp m2 1.13×10−2 9.55×10−3
Volume Vp m3 2.38×10−5 2.00×10−5
Density rp kg m−3 1.25×103 1.25×103
Moment of inertia Ip kg m2 1.2×10−4 1.2×10−4
f1 m4 5.0×10−5 7.2×10−5
f2 m5 3.8×10−6 5.7×10−6
f3 m4 2.4×10−5 2.4×10−5Shape factor
f4 m5 2.0×10−6 1.9×10−6
f5 m4 5.6×10−5 3.1×10−5
f6 m5 3.5×10−6 2.1×10−6
Variable Symbol Units Values
WeightMass mw kg 9.4×10−3
Position l1 m 2.5×10−2
Position l2 m 3.7×10−2
Height l3 m 2.0×10−2
lw=1/2√(l1+l2)2+l32 m 3.3×10−2
p=tan−1[l3/(l1+l2)] rad 0.31Thickness tw m 3.4×10−3
Volume Vw m3 8.2×10−7
Density rw kg m−3 1.15×104
Moment of inertia Iw kg m2 1.0×10−5
Table 2. Test cases
Non-dimensional Initial openingdistance, d angle, a0
0.020.060.09 0, p/20.19`
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The two plates A and B were designed to have almost the same values in these terms atevery opening angle a. The common relationship between a and Mf was obtained,therefore, for both plates A and B and is shown in Fig. 10B and Fig. 11B.
Flow visualization
Edwards and Cheng (1982) and Wu and Hu-chen (1984) pointed out that forcesacting on plates in ‘near fling’ motion are strongly influenced by vortices shed fromthe outer edges of plates. In order to make clear the structure of the vortex systemaround a plate in ‘near fling’, two kinds of flow visualization tests, three-dimensionaland two-dimensional, were performed. The three-dimensional test is more realistic insimulating the flow around an insect wing. The two-dimensional test is effective forunderstanding the fundamental structure of the vortex system. The two-dimensionaltest also provided results that could be compared with results of previous studies,which used two-dimensional models (Ellington, 1984a; Spedding and Maxworthy,1 9 8 6 ) .
Plates of two different shapes, plate A and plate B, were used in the three-dimensionaltest. Two different distances between the plate and the mirror, d=0.02 and d=`, were set
223Analysis of ‘near fling’
Mirror
L
Load cell
MY
a
FZ
Mf
FX
p
mwlwa2
mwlwaFS
mpxCGa2
FN +mpxCGa
d0
Fig. 3. Geometrical arrangements of the plate and the mirror and the definitions of forces andmoments.
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for each plate shape. The experimental apparatus is shown in Fig. 1. The pictures of flowwere taken by two still cameras, cameras 1 and 2. The wake was made visible by use ofthe dye Methylene Blue (Sunada et al. 1989).
Fig. 4A,B shows photographs for the cases of distance d=0.02 and d=`, respectively,for plate A with the initial opening angle a0=0. The photographs taken by camera 1 andby camera 2 show the flow pattern at a≈2 and a≈0.7, respectively. There is littledifference of vortex systems between the two distances, d=0.02 and d=`. A sketch ofthese vortex systems is shown in Fig. 5A. It can be seen that the vortices are generatedfrom all four edges. The vortices from the edge indicated by the shaded area in Fig. 5Amoved together with the plate and were always located near the plate surface. Thevortices from the other edges were left in the water. It is estimated that the vortex near therotation axis, the inner vortex, disappears. This is because it is cancelled by the innervortex from the other (e.g. the image) plate when the distance d becomes zero. One vortexring is, then, generated from the pair of rectangular plates.
Fig. 4C,D shows the results for the cases of distance d=0.02 and d=`, respectively, forplate B. There is little difference in the vortex systems at the two distances, d=0.02 andd=`. A sketch of these vortex systems is shown in Fig.5B. Again, it is observed that thevortices generated from the inner edge indicated by the shaded area in Fig. 5B movetogether with the plate. The vortices from the outer edge are left in the water. When thedistance d becomes zero, two vortex rings are generated and they are linked to each otherat the connecting point of the two triangular plates.
The flow pattern in the two-dimensional test was made visible by the use of aluminiumdust floating on the water surface. The experimental apparatus is shown in Fig. 6A. A testplate, the chord length of which is 60mm, rotates around an axis which is perpendicularto the water surface. The non-dimensional distance between the rotation axis and the wallis d=d0/cv=0.4. Pictures of the flow pattern were taken by a still camera set above thewater tank. The rotation speed V was about p/2rad s−1 and the Reynolds number,Re=0.75cv2V/n, was 3×103. The result is indicated in Fig. 6B. The vortices from an inneredge were always located near the surface of plate, but the vortices from an outer edgewere left in the water. This behaviour of vortices was commonly observed in the results ofthe three-dimensional test (Fig. 5A,B). The same behaviour of vortices from the outeredge was also observed in the results of two-dimensional ‘fling’ (Spedding andMaxworthy, 1986). The behaviour of the vortices is different, however, from that in thesketch of two-dimensional ‘near fling’ by Ellington (Fig. 4c of Ellington, 1984a), wherethe inner vortices were left in the water and the outer vortices were located near the platesurface.
Numerical calculation
A model was developed to calculate the entire time histories of the normal force andmoment acting on a pair of three-dimensional triangular plates rotating symmetricallyaround an axis. The normal force is drag because it is parallel to the inflow velocity due tothe plate rotation.
It is assumed that the pressure fields around the plates can be predicted by the use of the
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potential flow method. The velocity potential f of the flow field around a pair of platessatisfies the following Laplace equation:
£2f =0. (4)
As in Theodorsen’s method (Theodorsen, 1934), this velocity potential is divided intotwo components, a non-circulatory component fNC and a circulatory component fC asfollows:
f = fNC + fC . (5)
The non-circulatory component satisfies the boundary condition on the surfaces of theplates by using bound vortices and without vortices shed into the flow. The sum of thecirculation of these bound vortices is zero. Therefore, this component of the velocitypotential was expressed by using sinks and sources instead of bound vortices inTheodorsen’s method, and it was called a ‘non-circulatory’ component. The circulatorycomponent is generated by the vortices shed into the flow and does not change theboundary condition on the surfaces of the plates. The non-circulatory component satisfiesthe Laplace equation, that is:
£2fNC =0. (6)
The solution of the above equation is obtained by using a vortex lattice method (Levin,1984; Katz, 1984). A plate is composed of small panels as shown in Fig. 7A. The ordersof panels in the x- and y-directions are represented by j and k, respectively, as shown inFig. 7B. The mth panel is defined as:
m = kmax(j − 1)+ k . (7)
This panel has the mth non-circulatory component of bound vortex, GNC,m,ic and the mt hcollocation point is placed at the centre of the panel. The direction of circulation of the
225Analysis of ‘near fling’
Vortex
Plate
Rotation axis
Vortex
Plate
Rotation axis
BA
Fig. 5. Sketch of flow pattern of three-dimensional plates. (A) Flow pattern for plate A;(B) flow pattern for plate B.
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bound vortex is that of the right-handed revolution along the arrow shown in Fig. 7B. Thesuffix ic represents the current time step. The following equation is derived with the use ofthe 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 generatedby the bound vortices of a pair of plates. The influence coefficient matrices [An , m] and[Bn , m] represent the induced velocity on the nth collocation point generated by the unitnon-circulatory component of bound vortices of the mth panels on the same plate and theother plate, respectively. Hence, [Bn , m] depends on the opening angle a and the distance d.When an isolated plate is considered, [Bn , m] becomes the zero matrix. The second termrepresents the moving velocity of each collocation point, and the matrix [Dn ,1] is defined asthe distance between the collocation point and the rotation axis. By solving equation 8 ateach time step, the non-circulatory component of the bound vortex GNC,m,ic is obtained.
The total velocity potential, which is the sum of the circulatory component and the non-circulatory component, is obtained by solving equation 4. Like the model for the non-circulatory component, the mth panel has the mth bound vortex Gm,ic as shown inFig.7C. As confirmed by the results of the flow visualization test, no bound vortex issettled 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 flowvisualization 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 theouter edges, however, so the effect of the inner vortices on the pressure distribution of theplate is less than that of the outer vortices. The vortices are, then, represented by vorticesgenerated 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 thesurface of the cylinder swept by the edge BC, as shown by Fig. 7D. This surface of thecylinder, which is shaded in Fig. 7D, is extended to a plate, as shown in Fig. 7E. Thewake 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. Thestrength of this vortex sheet is defined by the circulation at the four corners of theelement, Gkmax(jmax−1)+k,i, Gkmax(jmax−1)+k,i−1, Gkmax(jmax−1)+k−1,i, Gkmax(jmax−1)+k−1,i−1, as shown inF i g . 7E. The method used to calculate the velocity induced by a rectangular vortex sheet inthe wake element is given by Johnson (1980). The following equation is derived from theboundary condition that the flow moves along the plate surface at each collocation point:
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228 S. SUNADA AND OTHERS
B
d Collocation points
+ Points where forceis calculated
wj,k
C
DSj,k
DlS,j,k
A y
DlT,j,k
k
j
x
B C
D
A
C
A
Vortex sheet
Gm,ic=G(j=j,k=k)
G(j=jmax−1,k=k)
G(j=jmax,k=k)
C
G(j=1,k=k)Collocation point
B
GNC(j=1,k=k)
GNC,m,ic=GNC(j=j,k=k)
GNC(j=jmax−1,k=k)
GNC(j=jmax,k=k)
B C
C
Collocation point
A
Rotation axisPlate at t=t
Plate at t=t
Plate at t=0Vortex sheet
C′C
BAA′B′
Fig. 7
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Again, the first term represents the induced velocity at the collocation point generated bythe bound vortices of a pair of plates. The second term represents the moving velocity ofeach collocation point. The third term represents the induced velocity of the collocationpoint generated by the vortices shed into the flow. By solving equation 9 at each time step,the circulation of the bound vortex Gm,ic is obtained. Because the non-circulatorycomponent of the mth bound vortex was obtained using equation 8, the circulatorycomponent of mth bound vortex is then given by:
GC,m,ic =Gm,ic −GNC,m,ic . ( 1 0 )
229Analysis of ‘near fling’
E A
B C
Plate
Rotation axis
Vortex sheet
C′B′
A′ Rotation axis
Plate
k=1
k=0k=kmax+1
k=kmax
i=ic
i=ic−1
i=1
i=0
i=0
i=1
i=2
i=ic
i=ic−1
Gkmax(jmax−1)+k,i
Gkmax(jmax−1)+k−1,i
Gkmax(jmax−1)+k−1,i−1
Gkmax(jmax−1)+k,i−1
Fig. 7. Model for numerical calculation. (A) Panel shapes on a plate and arrangement ofcollocation points and points where force is calculated; (B) arrangement of non-circulatorybound vortex GNC; (C) arrangement of total bound vortex G; (D) wake geometry assumed innumerical calculation; (E) circulation of separated vortices.
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230 S. SUNADA AND OTHERS
The total normal force FN,icand total moment around rotation axis Mf,ic are given by:
where
The first terms, FN1,ic and Mf1,ic, and the second terms, FN2,ic and Mf2,ic, in equation 11 aredue to dynamic pressure and impulsive pressure of the unsteady Bernoulli’s equation,respectively (Lamb, 1932). The quantities p and q in the suffix (p, q) of FN1(p,q),ic andMf1(p,q),ic in equation 12 are taken to be either a non-circulatory component NC or acirculatory component C. The quantity p indicates the type of vortex component thatgenerates the induced velocity at the point of the bound vortex. The quantity q indicatesthe type of component of the bound vortex. When an isolated plate is considered, theinduced velocity at the point of the bound vortex that is generated by the non-circulatorycomponent is always perpendicular to the plate surface. This is because the non-
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circulatory component of the vortex is distributed only on the plate surface. Thefollowing components of normal force and moment for an isolated plate are thereforegiven by:
Calculation of added mass and added moment of inertia
The normal force and the moment around rotation axis at t≈0 are mainly due toimpulsive pressure of the non-circulatory component. The difference in the non-circulatory component of velocity potential between the upper and lower surfaces DfNC
is proportional to the angular velocity a.. Therefore, DfNC is expressed as:
DfNC = a.G(a). (14)
When an isolated plate is considered, G is a constant instead of a function of openingangle a. The pressure difference distributed on the plate due to the non-circulatorycomponent is given by:
The non-circulatory components of the normal force and the moment around the rotationaxis acting on the plate are obtained by integrating the pressure difference on the platesurface as:
where
As shown in the above equations, the non-circulatory component of the normal force (themoment) is composed of two components. One is proportional to the angular accelerationa... The quantities VF and VM are shape factors; they are proportional to added mass and
added moment of inertia, respectively. The second component is proportional to a.2.
When both the distance d and the opening angle a are small, VF/a and VM/a arelarge, with the result that each of the second components is not small. This will be shownquantitatively later. The non-circulatory components of the normal force and the momentare proportional to angular acceleration a
..only at the very beginning of the motion t≈0.
This is because the angular velocity a.
is nearly 0rad s−1 at this moment, and the second
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components of the force and the moment are negligible. The shape factors VF and VM can,therefore, be obtained from values of FN(t=0), Mf(t=0) and a
..(t=0) by using the following
equations:
Normal force FN(t=0) and moment Mf(t=0) acting on a plate that begins to rotate atangular acceleration a
..(t=0) are obtained by using the present numerical calculation
method and from experimental data. The shape factors can then be given by both thenumerical and experimental methods.
Results
Fig.8A–D shows the motion of plates that occurred when the initial opening angle a0
was 0. Time histories of opening angle a for plates A and B are shown in Fig. 8A and B,respectively. Fig. 8C and D indicate, respectively, how a
.and a
..varied with a for two
232 S. SUNADA AND OTHERS
Fig. 8. Time histories of plate motion (a0=0). (A) Opening angle a of plate A; (B) openingangle a of plate B; (C) angular velocity a.; (D) angular acceleration a.
4
1
0.5
0
1
0
−1
2
0
4
2
0
d=0.02
d=`
d=0.02
Plate B,d=`
Plate B, d=`
Plate A, d=0.02
Plate A, d=0.02
d=`
d=0.06d=0.09
d=0.19
d=0.06d=0.09d=0.19
0 0.25p 0.5p 0.75p p 0 0.25p 0.5p 0.75p p
0 0.25p 0.5p 0.75p p0 0.25p 0.5p 0.75p p
Opening angle, a (rad)
B
C
D
A
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specific cases. The results for other cases are located between these lines. It is observedthat the variation of the angular acceleration is large at the beginning of the motion andthat the results of the angular velocity for each of the cases are similar to each other atabout a
..=0rad s−2.
Fig. 9A–C shows the motion of plates with a0=p/2 for two specific cases. It isobserved that there is little difference among all the cases at t=0.
The measured and calculated forces and moments acting on a plate are indicated inFig. 10A–D for a0=0 and in Fig. 11A,B for a0=p/2. The shaded areas in Fig. 10A andFig. 11A show the normal force FN and the tangential force FS, which were calculatedwith equation 1 using the measured values of FX and FZ obtained from the load cell. Theresults of all the experimental cases are located within this shaded area. It is observed thatthe difference is small among the cases. The bold lines in Fig. 10B and Fig. 11B show theexperimental results of the relationship between a and Mf, which is the same for bothplate A and plate B, as stated before. The solid lines in Fig. 10A,B and Fig. 11A,B showthe analytical results for test cases with d=0.02, and the broken lines in Fig. 10A,B showthe analytical results with d=`. The total values of the normal force and the moment, FN
and Mf, are compared with the components, FN1 and Mf1, which are caused by thedynamic pressure. It is observed that the total values obtained by calculation are in goodagreement with the experimental results for both cases. Therefore, it is verified that thiscalculation technique provides a means of estimating the normal force and the momentaround the rotation axis, which act on a triangular plate during ‘near fling’ with variousdistances d.
The chain lines in Fig. 10A,B and Fig. 11A,B show the results obtained by thenumerical calculation, where the measured plate movement for d=0.02 is assumed andthe induced velocity generated by the other plate is ignored. A comparison of the chainlines with the solid lines reveals the interference effect between the two plates of a pair. Itis observed that this interference effect is very small when the opening angle a is nearp/2.
Fig.10C,D shows the circulation of the bound vortex which is generated alongy=−ctip/4 at a=0.3, and each component of the normal force indicated in equation 12. Inthese figures, the solid, dotted and broken lines indicate the components of the boundvortex 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 withd=0.02. The narrow lines express the results obtained by using the calculation model,where the measured plate movement for d=0.02 is assumed and the induced velocitygenerated 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. Itis observed that this interference increases the values of GC near the rotation axis. Becausethe timewise change of integrated GC along the x-axis is equal to the circulation of thevortices shed into the flow, the interference increases this circulation. The inducedvelocity component along the surface of the plate is made larger by the strongercirculation of both the vortices generated from the plate and GC by the other (e.g. theimage) plate. The increment of the induced velocity component is suppressed by thestronger circulation of the vortices generated from the other plate. As a result, the induced
233Analysis of ‘near fling’
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234 S. SUNADA AND OTHERS
B
C
2
1
0
0
−2
0
2
4
A
0.5p 0.75p p
Opening angle, a (rad)
Plate A, d=0.02
Plate B, d=`
Plate A, d=`
Plate B,d=0.02
1
0.5
0.5p 0.75p p
Plate A, d=0.02
Plate B, d=`
0.5p 0.75p p
Plate A, d=0.02
Plate B, d=`
Fig. 9
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velocity component along the surface of the plate is made larger by the interferencebetween plates of a pair. The bound vortex circulation G near the rotation axis on the plateis also increased by the interference. Hence, the value of FN1(C , C)+FN1(C , N C) is increased,because the value of FN1(C , C)+FN1(C , N C) is proportional to the multiple of the bound vortexcirculation G and the induced velocity component along the plate surface. The othercomponent of normal force, FN1(N C , N C), the value of which is zero for an isolated plate asindicated in equation 13, has a positive value. Therefore, the interference increases thenormal force due to dynamic pressure, FN1(C , C)+FN1(C , N C)+FN1(N C , N C), and the momentaround the rotation axis, Mf1(C , C)+Mf1(C , N C)+Mf1(N C , N C). This is indicated as the difference ata=0.3 between the solid line and the chain line in Fig. 10A,B. When the opening angle aincreases, this interference between the two plates of a pair becomes small. It is alsoobserved in Fig. 10D that the value of FN1(C , C) is dominant when the angular velocity a
.
becomes large.Both the measured and the calculated added mass and added moment of inertia of the
rotating plate are shown in Fig. 12A–E. Fig. 12A,B shows the variations of calculatedshape factor VF and calculated position of centre of pressure with distance d. The resultsfor two different plates, plate A and plate B, and two different opening angles, a=0 anda=p/2, are shown. When the opening angle a is zero, the added mass, which isproportional to the shape factor VF, increases greatly as the distance d becomes small.This is because the fluid between the two plates is pulled by both the plates. The fluidbetween 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 nF i g . 12B that the position of the centre of pressure is not changed when the added massincreases. Therefore, the added moment of inertia increases in proportion to the increasein the added mass. In contrast, the added mass and the position of the centre of pressureare almost constant for the various distances d with a=p/2. The ratio of the measured tothe calculated added masses and the ratio of the measured to the calculated addedmoments of inertia are shown in Fig. 12C and D, respectively. The results indicate goodagreement between measurement and calculation. Fig. 12E shows the variation of addedmass with the opening angle a for two different distances d. It is observed that the addedmass decreases significantly as the opening angle increases, especially in the case of asmall distance d. This large variation of the added mass with opening angle gives theconsiderable values of a
.2(VF/ a) and a.2(VM/ a) in equation 16. The normal force due
to the impulsive pressure of the non-circulatory component and each of its componentsgiven by equation 16 are shown in Fig. 13 with d=0.02. The chain line indicates thec a l c u l a t e d FN2(N C), where the measured plate movement for d=0.02 is assumed andwhere the induced velocity generated by the other plate is ignored. The differencebetween the solid line and the chain line indicates the effect of interference between twoplates. This interference increases the normal force due to the impulsive pressure of thenon-circulatory component for the period 0<a<0.1 only. The component a
.2(VF/a)reduces the total value of the normal force due to the impulsive pressure of the non-
235Analysis of ‘near fling’
Fig. 9. Time histories of plate motion (a0=p/2). (A) Opening angle a; (B) angular velocity a.;(C) angular acceleration a.
jeb9010.q 20/11/98 12:22 pm Page 235
236 S. SUNADA AND OTHERS
B
C
1
0.5
0
−0.5
A
Experiment, FS
Opening angle, a (rad)
Experiment, FN
FN2
FN1
0 0.25p 0.5p 0.75p p
5
0
Opening angle, a (rad)
Experiment
Mf2
Mf1
0 0.25p 0.5p 0.75p p
1
0
−10 0.5 0.75 1.00.25
GC
G
GNC
Position along y=−ctip/4, 2x/xtip−1
Fig. 10
jeb9010.q 20/11/98 12:22 pm Page 236
circulatory component, and this total value with the interference is smaller than thatwithout the interference during the period a>0.1. In summary, the beating motion, whereboth the distance d and the opening angle a are small, makes it possible to increase theadded mass of a wing. The normal force due to the impulsive pressure of the non-circulatory component, however, is increased during the very limited period in which theopening angle is very small.
Discussion
Capability of simple analysis
A simple model was proposed for analyzing the wing-beating flight of insects (Weis-Fogh, 1973; Ellington, 1984b). This simple model and another simple model newlypresented in this paper are compared with the present experimental and numericalcalculations in order to make clear the capability of these simple models. The previoussimple model by Weis-Fogh or Ellington used a plate element parallel to the rotation axis(plate element 1) as shown in Fig. 2, and the forces acting on this element wereconsidered. The present simple model used a plate element perpendicular to the rotationaxis (plate element 2). The calculated results are compared with the experimental resultsfor the single-plate case (i.e. d=`).
Two typical instants, t=0 and a..=0rad s−2, are selected. The force and moment acting
237Analysis of ‘near fling’
5
0
−2.50 0.25p 0.5p 0.75p p
D FN1(C,NC)+FN1(C,C)
FN2(NC)
FN1(C,C)
FN1(C,NC)FN1(NC,NC)
FN2(C)
Opening angle, a (rad)
Fig. 10. Time histories of normal and tangential forces and moment around rotation axis(a0=0). (A) Normal force, FN and tangential force, FS. (B) Moment around rotation axis, Mf.Hatched area or ——, experimental results; ——, calculation result with d=0.02; – – –,calculation result with d=`; — - —, calculation result where the effect of the other (e.g. theimage) plate is ignored with d=0.02. (C) Circulation distribution of bound vortex on y=−ctip/4along the x-axis at a=0.3rad. (D) Component of normal force. Bold lines, calculation resultwith d=0.02; narrow lines, calculation result where the effect of the other plate is ignored withd=0.02.
jeb9010.q 20/11/98 12:22 pm Page 237
on a plate are caused by the added mass alone at t=0, and by the ‘quasi-steady’ force aloneat a
..=0rad 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 secondpower of angular velocity of the plate.
Calculation at t=0
It is assumed that a plate element has an added mass equal to the two-dimensional plate
238 S. SUNADA AND OTHERS
1
0.5
0
−0.5
A
Experiment, FS
Experiment, FN
FN2
FN1
0.5p 0.75p p
B
5
0
Opening angle, a (rad)
Experiment
Mf2
Mf1
0.5p 0.75p p
Fig. 11. Time histories of normal and tangential forces and moment around rotation axis(a0=p/2). (A) Normal force, FN and tangential force, FS. (B) Moment around rotation axis, Mf.Hatched area or ——, experimental results; ——, calculation result with d=0.02; — - —,calculation result where the effect of the other (e.g. the image) plate is ignored with d=0.02.
jeb9010.q 20/11/98 12:22 pm Page 238
of the same width c(x). A plate element 1, then, has the following added mass (Lamb,1932):
dmf = prf{c(x)/2}2dx . (19)
The normal force FN(t=0) and the moment around the rotation axis Mf (t=0), which arecaused by the angular acceleration of the plate at the beginning of motion, are obtained byintegrating their components acting on the plate element 1 as follows:
The shape factors, f1 and f2, are defined as:
The non-dimensional shape factors are given by:
whereR = ctip/xtip . (25)
As in the previous simple model of Weis-Fogh or Ellington, plate element 2 is assumedto be part of the two-dimensional plate of the same width b(y) rotating with angularacceleration a
... The normal force and the moment around the rotation axis generated on
this element are given by (Lamb, 1932):
where
239Analysis of ‘near fling’
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240 S. SUNADA AND OTHERS
B
C
5
0
1
1
0 0
Plate A
Plate B
Plate A
Plate B
a=0
a=p/2
A
0 0.1 0.2 `
Non-dimensional distance, d
Plate A
Plate B
Plate A
Plate B
a=0
a=p/2
0 0.1 0.2 `
1
0
Plate APlate B
Plate A
Plate B
a=0
a=p/2
0 0.1 0.2 `
2
1
0
Fig. 12
jeb9010.q 20/11/98 12:23 pm Page 240
The normal force and moment around the rotation axis at t=0 are given by integratingthese compoments of plate element 2 as follows:
241Analysis of ‘near fling’
Plate A
Plate B
Plate A
Plate B
a=0
a=p/2
D
Non-dimensional distance, d
Opening angle, a (rad)
0 0.1 0.2 `
2
1
0
5
0
1
1
0 0
d=0.02
d=0.09
E
0 0.5 1.0 1.5
Plate A
Plate B
Plate A
Plate B
Fig. 12. Added mass of plate A and plate B. (A) Shape factor VF versus non-dimensionaldistance d; (B) centre of pressure VM/VFxtip versus d; (C) ratio between measured andcalculated shape factor VF; (D) ratio between measured and calculated shape factor VM; (E)shape factor VF versus opening angle a.
jeb9010.q 20/11/98 12:23 pm Page 241
where
The non-dimensional shape factors are given by:
Comparing equations 20 and 28 with equation 18, it should be noted that shape factor f1or f5 corresponds to shape factor VF and that shape factor f2 or f6 corresponds to shapefactor VM. Shape factors VF and VM are dependent not only on plate shape but also onopening angle a and distance d, because the interference effect between a pair of platescan be considered for calculating these factors by using the vortex lattice method. In
242 S. SUNADA AND OTHERS
1
−1
0
0 0.5
Opening angle, a (rad)
rfa2(VF/a)
FN2(NC)≈rf aVF,rf aVF
FN2(NC)=rf {aVF+a2(VF/a)}
without the effect of the other plate
Fig. 13. Normal force due to impulsive pressure of non-circulatory component. ——, normalforce with d=0.02, FN2(NC)=rf{aVF+a
.2(VF/a)}; – – –, normal force component withd=0.02, rfaVF; —--—, normal force component with d=0.02, rfa
.2(VF/a); — - —, normalforce where the induced velocity generated by the other (e.g. the image) plate is ignored withd=0.02, FN2(NC)≈rfaVF.
jeb9010.q 20/11/98 12:23 pm Page 242
contrast, shape factors calculated by the simple methods, f1, f2, f5 and f6, are dependent onplate shape alone.
F i g . 14A–D shows the variations of the non-dimensional shape factors with thevariation of the non-dimensional plate geometry ratio R=ct i p/xt i p. The results for the twosimple models, obtained by use of equations 23, 24, 31 and 32, are compared with theresults obtained by using the present vortex lattice method. It is observed that theprevious simple model approach (Weis-Fogh, 1973; Ellington, 1984b) is in goodagreement with the vortex lattice method when the ratio R is less than 0.5. In contrast,the present simple model approach provides acceptable solutions when the ratio R i slarge (R>2). This means that the flow around the plate at the beginning of the platemotion is parallel to plate element 1 for small values of R and to plate element 2 for largevalues of R near the beginning of the plate motion, t≈0. It is also observed that neither ofthe simple model approaches provides acceptable predictions for an intermediate valueof the ratio R (≈1), which corresponds to that for the so-called ‘low aspect ratio wing’such as that of a butterfly. In addition, it should be noted that, in using either of thesimple models, it is not possible to take into account the interference between the twowings of a pair. Therefore, the present vortex lattice method should be used to analyze
4; (B) non-dimensional shape factors of rectangular plate,f2/xtip
5, f6/xtip5 and VM/xtip
5; (C) non-dimensional shape factors of triangular plate, f1/xtip4,
f5/xtip4 and VF/xtip
4; (D) non-dimensional shape factors of triangular plate, f2/xtip5, f6/xtip
5 andVM/xtip
5.
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precisely a low aspect ratio wing such as a butterfly wing or the interference betweentwo beating wings.
Calculation at a=0 (‘quasi-steady’ hydrodynamic force)
The previous simple model (Weis-Fogh, 1973) estimates the ‘quasi-steady’hydrodynamic force at the instant a
..=0rad s−2 as follows. It is assumed that the velocity
induced by the vortices in the wake is ignored when counting the inflow velocity towardsplate element 1; that is, the relative velocity of the element is defined by the angularvelocity a
.alone. In addition, the ‘quasi-steady’ force coefficient of the element is
assumed to be kF,a. for calculation of the normal force or kM,a
. for calculation of themoment around the rotation axis, and both the coefficients are assumed to be constant forall elements. The normal force FN(a
..=0) and the moment around the rotation axis
Mf(a..=0) are expressed by integrating these components acting on the plate element 1 as
follows:
where
The values of kF,a. and kM,a
. are defined by substituting the measured values of force andmoment around the rotation axis for FN and Mf in equations 33 and 34. The results areshown in Table 3. It is indicated that these values are independent of the distance d. Thismeans that the interference between two plates of a pair has a negligible effect on thevalues of kF,a
. and kM,a. at a
..=0rad s−2, that is, a≈p/2. Differences in the values of kF,a
. orkM,a
. are observed in those cases where a0 is different, even though the Reynolds number
is the same. These differences are caused by the induced velocity generated by thevortices in the wake. In addition, these values of kF,a
. and kM,a. are much greater than the
steady drag coefficient, 1.98, acting on a two-dimensional plate in the normal flow at aReynolds number of 104 (Hoerner, 1958). This inconsistency observed in the calculatedvalues of kF,a
. and kM,a. shows the limitation of the applicability of the previous simple
model (Weis-Fogh, 1973) and indicates that the flow around the rotating plate is verydifferent from the assumed two-dimensional flow. However, this simple method is usefulin providing rough estimates of the hydrodynamic force generated by insect wings, if themodified hydrodynamic force coefficient (e.g. 4) is adopted.
Conclusion
The following characteristics of ‘near fling’ are made clear by the experiments and thenumerical calculation. The hydrodynamic force acting on a pair of rotating plates isperpendicular to each of the plate surfaces, and this force is drag because the force isparallel to the inflow velocity due to rotation of the plate. The major part of thehydrodynamic force is composed of two components: that due to the dynamic pressureand that due to the added mass.
The interference between each wing of a pair increases the added mass and thehydrodynamic force due to the dynamic pressure with a small opening angle a and asmall distance d between the wings. The hydrodynamic force due to the dynamic pressureis proportional to the bound vortex circulation and to the induced velocity componentalong the wing surface. The interference between two wings increases the bound vortexcirculation near the rotation axis and the induced velocity component along the surface ofone wing generated by the vortices shed from the wing and by the bound vortices of theother 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 isbalanced with the gravity force during insect flight, is roughly perpendicular to thehydrodynamic force when the opening angle is small. Therefore, the interference effectbetween a pair of wings is important for insect flight only when the angular velocity orangular acceleration of the flapping angle at a small opening angle is much larger than at alarge opening angle.
Three calculation methods, a vortex lattice method and two simple methods, are usedto analyze the hydrodynamic force. The newly developed vortex lattice method, in whichthe plate is represented by the vortex lattice, predicts well every experimental result. Thismethod has the capability of predicting both the interference effect between a pair ofwings and the effect of the vortices shed into the flow.
Both the simple methods use a plate element parallel to the rotation axis (plate element1) or perpendicular to the rotation axis (plate element 2). The simple method using plateelement 1 can estimate the added mass of isolated beating wings when the non-dimensional plate geometry ratio R is less than 0.5. This method, therefore, is effectivefor calculating the added mass of an insect’s wing with a small geometry ratio, such as adragonfly’s wing, when the interference effect between a pair of wings is negligible. Incontrast, the simple method using plate element 2 can estimate the added mass of isolated
245Analysis of ‘near fling’
jeb9010.q 20/11/98 12:24 pm Page 245
wings when the geometry ratio is greater than 2. The geometry ratio of butterfly wings isabout 1, and the simple method using either element fails to estimate its added mass. Inaddition, the simple method using either element fails to estimate added mass of anywings when there is an interference effect between a pair of wings.
The quasi-steady force coefficient, which has been used for the simple method usingthe plate element 1 (Weis-Fogh, 1973; Ellington, 1984b) and is defined by equations 33and 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 forceobserved in insect flight (Dudley, 1991).
The above results will be useful for understanding the ‘near fling’ elements of insectflight. The previous simple methods used for analyzing beating wings should beimproved to allow a quantitative analysis of ‘near fling’.
Appendix
It is shown in this section that the exact experimental method (method I) where twoplates 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 willgive the same results when the potential flow is assumed. The friction between the flowand the mirror, however, restricts flow along the mirror surface in method II. This createsthe possibility of a difference between the results of the two methods.
246 S. SUNADA AND OTHERS
Fig. 15. Experimental arrangement for method I.
Plate
2FZ
Plate
2d0
Load cell
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Fig. 15 shows the arrangement of the experimental apparatus for method I. The force indirection Z, 2FZ in equation 1, is measured by a load cell. It is predicted that the differencebetween the two methods is larger when the distance d is smaller and when therectangular plate (plate A) is used. Therefore, plate A, with the smallest distance inTable 2, d=0.02, is selected for this experiment.
The solid and broken lines in Fig. 16 show time histories of the force in direction Z, FZ,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 resultobtained by method II shown by a short broken line in Fig. 8A. Therefore, it is verifiedthat method I is equivalent to method II in the present research. Method II was usedbecause of the ease with which the experiment could be conducted.
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0.5
00 5
Method IMethod II
Time, t (s)
Fig. 16. Comparison of vertical forces measured using the two different methods.
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