Lagrangian Coherent Structures in Tandem Flapping Wing Hovering

Journal of Bionic Engineering 14 (2017) 307–316

Lagrangian Coherent Structures in Tandem Flapping Wing Hovering Srinidhi Nagarada Gadde, Sankaranarayanan Vengadesan Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai, Tamil Nadu, India

Abstract Lagrangian Coherent Structures (LCS) of tandem wings hovering in an inclined stroke plane is studied using ImmersedBoundary Method (IBM) by solving two dimensional (2D) incompressible Navier-Stokes equations. Coherent structures responsible for the force variation are visualized by calculating Finite Time Lyapunov Exponents (FTLE), and vorticity contours. LCS is effective in determining the vortex boundaries, flow separation, and the wing-vortex interactions accurately. The effects of inter-wing distance and phase difference on the force generation are studied. Results show that in-phase stroking generates maximum vertical force and counter-stroking generates the least vertical force. In-phase stroking generates a wake with swirl, and counter stroking generates a wake with predominant vertical velocity. Counter stroking aids the stability of the body in hovering. As the hindwing operates in the wake of the forewing, due to the reduction in the effective Angle of Attack (AoA), the hindwing generates lesser force than that of a single flapping wing. Keywords: dragonfly, tandem wings, LCS, IBM Copyright © 2017, Jilin University. Published by Elsevier Limited and Science Press. All rights reserved. doi: 10.1016/S1672-6529(16)60399-2

1 Introduction Vortex dynamics and flow separation at high Angle of Attack (AoA) play a significant role in the aerodynamics of insects, birds, and Micro Aerial Vehicles (MAVs). Force generation in flapping flight is coupled to the existence of Leading Edge Vortex (LEV)[1], momentum transfer, and entrainment of surrounding fluid by counter-rotating vortices. As a result, the identification and the analysis of vortex structures become significant. Shyy and Liu[2] gave a comprehensive review of the aerodynamics of flapping wings. Most of the vortex feature identification techniques like Q and lambda criteria[3] require a user-defined, preselected threshold to define the boundaries of vortices[4]. By applying concepts of dynamical systems theory to fluid motions, Haller and Yuan[5] proposed the Finite-Time Lyapunov Exponent (FTLE) fields to visualize the Lagrangian Coherent Structures (LCS). Haller and Peacock[6] reviewed LCS and its application to geophysical flows. Based on the work of Haller[5,7], Shadden et al.[8] presented an improvised definition of LCS as a ridge of FTLE field. Deformation tensor of the fluid is calculated over a finite time interval, and the maximum eigenvalue Corresponding author: Sankaranarayanan Vengadesan E-mail: [email protected]

of the tensor represents the ridges of FTLE. LCS are the ridges of FTLE subjected to an additional hyperbolicity condition[9] to nullify the flux across the boundaries of the structures. FTLE obtained by integrating forward in time quantifies the separation between two nearby particles over the time interval. FTLE ridges, therefore, are curves along which particles are most prone to deviate from one another[10]. The ridges are called repelling LCS analogous to stable manifolds in dynamical systems theory. In contrast, if the integration is performed backward in time, then the ridges attract two particles which are separated by a distance in the beginning. The ridges are analogous to unstable manifolds and are called attracting LCS. In the past decades, LCS have been utilized to visualize vortex structures in diverse fields. Peng and Dabiri[11] investigated the wake dynamics of both swimming and flying animals by examining both repelling and attracting LCS. Brunton and Rowley[12] used LCS to visualize vortical structures in the wake of a flat plate undergoing pitching and plunging. Wan et al.[13] studied the wake vortices in a plate undergoing harmonic and non-harmonic pitching and plunging using backward FTLE ridges. Eldredge and Chong[10] studied fluid

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transport and coherent structures of translating and flapping wings, and compared the change in the vortical structures for both rigid and exible wings. Yang et al.[14] visualized the repelling and attracting LCS of a starting vortex ring generated by a thin circular disk. Their results revealed a flux window between the attracting and repelling structures which entrains the shear flow into a vortex. Recently, Rosti et al.[15] performed Direct Numerical Simulation (DNS) of the flow around a pitching airfoil at high Reynolds number and visualized Kelvin-Helmholtz instabilities by employing backward FTLE ridges. Dragon flies are one of the highly manoeuvrable flyers with independently moving fore- and hind wings. Lan and Sun[16] solved 2D incompressible Navier-Stokes equations on moving overset grids to study hovering elliptical foils at 0˚, 90˚, and 180˚ phase differences. Wang and Russell[17] proposed an idealized kinematics mimicking dragonfly kinematics and studied the power requirements in a dragonfly hovering. They found that counter-stroking utilizes minimal power and generates sufficient lift to keep an insect aloft. Specific power consumption in hovering reduces with elastic storage in the muscles (Shen and Sun[18]). Usherwood et al.[19] used robotic wing experiments to study the vorticity dynamics of a dragonfly flight and proved that a dragonfly employs wing phasing to remove swirl from the wake and improves the aerodynamic efficiency. Xiang et al.[20] performed a parametric analysis of corrugated tandem wings, and their results show that lift-drag ratio for the wings is only marginally affected by the corrugations. Broering and Lian[21] studied tan tandem wings pitching and plunging in a vertical plane, and showed that both inter-wing distance and wing phasing can be used to control the force generation. Most of the past studies have focused on tandem wings pitching and plunging in a vertical plane. Recently, Broering and Lian[22] extended the work to 3D and showed that at low Reynolds number, 2D simulations reasonably predict the unsteady mechanisms of force generation. However, for the LCS, the effects of inter-wing distance and wing phasing on force generation have not been studied for systems mimicking dragonfly kinematics. In this paper, we study a virtual tandem flapping wing model performing idealized dragonfly kinematics in a quiescent fluid. Velocity fields are obtained by solving incompressible 2D Navier-Stokes equation us-

ing immersed boundary method. Backward FTLE ridges are used in conjunction with vorticity contours to study the effect of phase difference and inter-wing distance on the aerodynamics.

2

Governing equations and the numerical method

2.1 Immersed boundary projection method Immersed boundary projection method proposed by Taira and Colonius[23] is used in the present study. Incompressible 2D Navier-Stokes equations are solved on a cartesian grid called Eulerian grid, D, and a set of discrete Lagrangian points, ξk, represent the surface of the body, B. The governing equations are:

∂u + u ⋅ ∇u = −∇p + νΔu + ∫ f (ξ ( s, t ))δ (ξ − x )ds, (1) S ∂t ∇ ⋅ u = 0,

(2)

u(ξ ( s, t )) = ∫ u( x )δ ( x − ξ )dx = uB (ξ ( s, t )), (3) S

where x ∈ D and ξ(s, t) ∈ B. The boundary B, is parameterized by s, and moves at the velocity, uB(ξ(s, t)). Staggered grid finite difference formulation is used to discretize the above equations with pressure at the center of the cell and velocity fluxes on the cell faces. Explicit second order Adams-Bashforth scheme is used to discretize the convective terms and implicit CrankNicholson scheme is used to discretize the viscous terms. The discretization yields a formal accuracy of second order in space and first order in time. 2.2 FTLE calculation If x(t) represents fluid particles initialized over the flow field at a time t, u(x,t) represents the time dependent velocity, and x(t + T) represents the position of the particles after a time T, the trajectory of a fluid particle is obtained by:

φtt +T ( x ) = x (t + T ) = x(t ) + ∫

t +T

t

u ( x, t )dt.

(4)

An in finitesimal separation, δx, at the initial time t changes to δx(t + T) by the relation:

δ x (t + T ) = ∇φtt +T ( x )δ x ,

(5)

where ∇φtt +T ( x ) represents the deformation gradient

Srinidhi and Vengadesan: Lagrangian Coherent Structures in Tandem Flapping Wing Hovering

tensor and is given by: ∇φtt +T ( x ) =

(6)

δ x (t + T ) = δ x[∇φtt +T ( x )]∗ ∇φtt +T ( x )δ x , where [∇φ

(7)

∗

( x )] represents the transpose of the de-

formation tensor ∇φtt +T ( x ) and Cauchy-Green deformation tensor: Δ = [∇φtt +T ( x )]∗ ∇φtt +T ( x ).

2 1

The magnitude of separation/stretching is given by:

t +T t

Xu et al.[26] Gao et al.[25] Present

3

dφtt +T ( x ) . dx

0 −1 85.0

87.5

λmax (Δ) gives the maximum separation/stretching, thus,

92.5 t

95.0

97.5

100.0

Fig. 1 Validation study of inclined stroke plane hovering.

In the present study, we consider tandem wings hovering in a quiescent fluid. We use the kinematics proposed by Wang[24], as explained in detail in Fig. 2. Kinematics of the wings is given by: [ x(t ), y (t )] =

(9)

FTLE is defined as:

σ tt +T

90.0

4 Problem formulation (8)

Let λmax(Δ) be the maximum eigenvalue of the CauchyGreen deformation tensor. Eq. (7) shows that

δ xmax ( x + T ) = λmax (Δ ) δ x .

309

1 1 δ x (t + T ) = ln λmax (Δ ) = ln . (10) T T δx

FTLE quantifies the amount of stretching/separation of fluid particles over a particular time interval, and the ridges of FTLE represent LCS. FTLE can be computed for both T > 0 and T < 0. The ridges are called attracting LCS when the integration in Eq. (4) is performed backward in time. The integration time T needs to be chosen based on the time scale of coherent structures in the flow. Larger integration time leads to visualization of more boundaries, and the use of smaller integration time leads to the revelation of less boundaries. Increasing the integration time gives rise to the visualization of finer structures.

3 Validation: Inclined stroke plane hovering In addition to validating the solver against standard benchmark cases, flow over a wing with elliptic cross section with chord length c, and minor to major axis ratio, 0.25, hovering in a quiescent fluid performing the kinematics proposed by Wang[24] is used to validate the solver. Fig. 1 shows the time varying vertical force coefficient, CV, plotted over two flapping cycles. The simulation results match well with the results from Gao et al.[25], and Xu et al.[26].

A0 cos(2πft + ψ )(cos β ,sin β ), (11) 2c

α (t ) = α 0 − α m sin(2πft + φ + ψ ), CH =

FH 1 ρU 2 c 2

CV =

FV 1 ρU 2 c 2

,

(12) (13)

where [x(t), y(t)] is the position of the center of chord of the wing, α(t) is the angle made by the chord with the stroke plane, β is the stroke plane angle, φ is the phase difference between translation and rotation, f is the frequency of flapping, A0/c (A0 is the distance travelled by the wing in a half stroke, and c is the length of the major chord of the ellipse.) and αm are the amplitudes of translation and rotation respectively. Velocity scale, U = π(A0/c)f, is related to oscillating translation. Reynolds number, Re = Uc/ν = πfA0c/ν, is based on the maximum velocity of translation and the chord length. T = 1/f is the time period of flapping. CH and CV represent the horizontal and vertical force coefficients respectively, and ψ is the phase difference between forewing flapping and hindwing flapping. The size of the computational domain is 20c × 20c. Three grids are considered, corresponding to uniform grid sizes of Δx, Δy = 0.02c, 0.01c, and 0.005c respectively. Total numbers of grid points in the three grids are shown in Table 1. Fig. 3 shows the time variation of vertical force for the three different grids. From Fig. 3 it is clear that grid-1 predicts the results with reasonable accuracy except for

Journal of Bionic Engineering (2017) Vol.14 No.2

310

importance of LCS in the studies of vortex structures.

∂u — = ∂v — =0 ∂y ∂y

β

l/c

A

0 /c

α

∂u ∂v =0 — =— ∂x ∂x

A0/c

∂v =0 ∂u — =— ∂x ∂x

∂u — = ∂v — ∂y ∂y =0 (b)

(a)

Fig. 2 (a) Kinematic positions of the wing in one flapping cycle. Continuous lines represent positions in downstroke and dashed lines represent positions in upstroke. β = π/3, A0/c = 2.5, α0 = αm = π/4, and Re = 100. (b) Computational domain and the boundary conditions. Forewing and hindwing are represented by white and gray ellipses respectively. The perpendicular distance between the stroke planes of the two wings is represented by l/c. Table 1 Grids considered in the study Grid-1

Grid-1

Grid-1

426 × 426

992 × 992

1526 × 1526

10 8

Grid-1 Grid-2 Grid-3

6

CV

4 2 0

5.1 Vorticity and LCS Fig. 4 shows the vorticity (Fig. 4a) and backward FTLE ridges (Fig. 4b) of wings flapping in tandem. To calculate FTLE, the velocity field is integrated over a time interval equal to twice the time period of flapping. The chosen integration time is sufficient to extract all the vortex boundaries, as the simulations are carried out at a low Reynolds number. Fig. 4 shows the correlation between vorticity and LCS. It is evident from Fig. 4b that LCS correlates well with the structures in vorticity contours (Fig. 4a). The thickest blue line in Fig. 4b represents the ridge along which a fluid element experiences maximum deformation. The lift generated by the wings depends on the size and strength of the shed counter-rotating vortex pair, as well as the size of the Leading Edge Vortex (LEV) on the wings. Attracting LCS shows that the fluid inside the spiral ridges entrains the surrounding fluid and transfers momentum. The thickest attracting LCS ridge in Fig. 4b represents the curve along which the deformation of the surrounding fluid element is maximum. As discerning the strength of vortices from the LCS is difficult, LCS needs to be used in conjunction with the vorticity contours to study the vorticity dynamics. In other words, the strength of the vorticity is given by the vorticity contours and intricate vortical structures are exposed by LCS.

−2 −4 79

80

81

82

t

83

84

85

86

Fig. 3 Grid independence study.

negligible over-predictions at the peaks. However, better resolution of the grids helps in visualizing high quality, and sharper LCS and vorticity contours. Since the difference between the results of grid-2 and grid-3 is negligible, we use grid-2 in the rest of our simulations.

5 Results and discussion We employ vorticity contours and backward FTLE ridges to explain the time-varying forces resulting from the vortex dynamics. We have used open-source flow visualization software VisIt[27] to compute the FTLE field from the velocity field data. We present the visualizations of hovering flapping wings to emphasize the

5.2 Force variation and vortical structures in tandem wing hovering

The parameters used in the simulations are: the perpendicular distance between the stroke planes of fore- and hindwings, l, and the phase difference between the flapping of forewing and hindwing, ψ. Seven inter-wing distances, l = 1.1c, 1.2c, 1.3c, 1.5c, 1.7c, 1.9c, and 2.1c and two phase differences ψ = 0° and 180° are considered in the study. The Reynolds number Re is 100 and the ratio of minor axis to major axis of the ellipse is 0.25. The time-variation of CV reaches a periodic state after four-five cycles of stroking in all the simulations. The forces are averaged over one flapping cycle after the tenth cycle. The time-averaged force coefficients are represented by CV and CH . As vertical force is the variable of prime importance in cases of hovering in quiescent fluid, we focus our discussions on the variation of CV.

Srinidhi and Vengadesan: Lagrangian Coherent Structures in Tandem Flapping Wing Hovering

311

Counter-rotating vortex pair

(a)

Ridge of maximum deformation

Vortex boundary (b)

Fig. 4 (a) Vorticity contours; (b) backward FTLE ridges (LCS). Blue represents clockwise (CW) and red represents counter clockwise (CCW) vorticity respectively.

Fig. 5 (a) and (b) Time varying CV of forewing and hindwing respectively; (c) effect of interwing distance on cycle averaged vertical force coefficient.

5.2.1 In-phase stroking, ψ = 0˚ In this section, force variation, the evolution of vortices, and the effect of inter-wing distance on the force variation of wings flapping in-phase are presented. Typical variation of CV with time and the corresponding evolution of vortices are explained for the case with inter-wing distance, l = 1.1c (Figs. 5 and 6). Figs. 5a and

5b show the variation of CV with time at different inter-wing distances. For the sake of comparison, the variation of CV for single wing flapping is also plotted (dotted line in Figs. 5a and 5b). From Fig. 5c, it is clear that forewing generates more force than a single flapping wing system, indicating that the presence of the hindwing enhances the force generation of the forewing.

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Fig. 6 Evolution of vorticity with time and LCS for l = 1.1c. Vorticity and LCS are plotted in alternating rows, subscripts F and H represent fore- and hindwings respectively.

In Figs. 5a and 5b, the initial peak in vertical force at t/T ≈ 0.05 is due to the reaction force provided by the fluid due to the acceleration of the wings and rapid pitch-down rotation of the wings[28]. Also the CCW wake vortex, represented by TEVP (Fig. 6a) interacts with the wing and transfers momentum. This interaction, the so-called wake capture, also enhances the force. As the wings continue their downstroke, CW LEV is developed at the leading edge of the wings in accordance with delayed stall mechanism (Fig. 6c). At t/T ≈ 0.12, unlike forewing, the CV of the hindwing starts increase, this is due to the interaction of the hindwing with the CCW trailing edge vorticity shed at the end of the previous upstroke of the forewing. This wing-wake interaction is the reason for the peak in CV of the hindwing at t/T ≈ 0.2 (Fig. 5b). TEVP is captured accurately in the LCS plot of Fig. 6a. At t/T ≈ 0.3, TEVP interacts with the hindwing, this corresponds to the local maxima in CV of the hindwing (Fig. 6b). LEVs and TEVs of the forewing and hindwing are represented by LEVF, TEVF, LEVH and TEVH respectively. At t/T ≈ 0.5, TEV of the forewing is shed (TEVF in Fig. 6c), and it interacts with the hindwing. As the wings start pitching up, due to the deceleration of the wings, CV reaches its minimum value. Shed TEV of the forewing merges with the TEV of the hindwing (TEVF + TEVH in Fig. 6d). The phenomenon of vortex merging is captured with finesse in the LCS contours. TEVF gets sheared, stretched and

ultimately merges with TEVH. As the wings continue with the upstroke, merged TEVs, and shed LEVs entrain surrounding fluid and transfer momentum. Jet created by the counter-rotating vortices form a part of the total vertical force in the upstroke and at the beginning of the downstroke. Time-averaged vertical force coefficient, CV , of the forewing and hindwing are 0.498 and 0.314 respectively (Fig. 5c). Hindwing constantly operates in the wake of the forewing; due to the effect of wake on the LEV generation, CV of the hindwing is less compared to the forewing. In downstroke, the presence of forewing is detrimental to the growth of the LEV of the hindwing. In upstroke, the hindwing is nearly vertical and it constantly moves in the downwash created by the forewing. The downwash increases the drag on the surface of the hindwing, consequently CV of the hindwing is less than that of a single wing flapping system. LCS in Fig. 6 shows the attracting dynamic structures in the flow which entrain the surrounding fluid. Time dependent behaviour of individual vortices like stretching and merging can be kept track with LCS (Fig. 6d). Fig. 5c shows the effect of inter-wing distance on the cycle averaged vertical force coefficient. CV of the hovering single wing is 0.442 (dashed line in Fig. 5c). Fig. 7 presents the vorticity contours at various inter-wing distances. As the inter-wing distance increases, the effect of the forewing on the LEV generation of the

Srinidhi and Vengadesan: Lagrangian Coherent Structures in Tandem Flapping Wing Hovering

(c) l = 1.7c

(b) l = 1.5c

(d) l = 1.9c

CV

Fig. 7 Vorticity contours for different l at the end of the downstroke. Strength of LEV and TEV increases with increasing inter-wing distance.

CV

5.2.2 Counter-stroking, ψ = 180˚ Figs. 8a and 8b represent the time variation of CV for different l, when the hindwing leads the forewing by 180˚. Fig. 8c shows the variation of time-averaged vertical force coefficient versus inter-wing distance. From Fig. 8a, it is clear that the presence of hindwing has only marginal effect on the CV of the forewing. Hindwing in its downstroke generates lesser force compared to a single flapping wing, as it operates in a low pressure area created by the shed TEV of the forewing. As the inter-wing distance increases, the effect of forewing on the force generation of hindwing decreases and consequently, for l > 1.5c the vertical force generation of the hindwing increases rapidly (Fig. 8c). Typical variation of CV is explained for l = 1.1c. Fig. 9 shows the evolution of vorticity and LCS contours with time. Hindwing is at the beginning of its upstroke when the forewing is at the beginning of its downstroke (LEVH in Fig. 9a). The initial peak in CV is reduced because of the presence of the LEV of the hindwing (Fig. 9a). At t/T = 0.3, the TEV of the forewing interacts with the CCW vorticity of the hindwing, and thus creating a low pressure region near the lower surface

(a) l = 1.3c

‒ CV

hindwing decreases and CV of the hindwing increases and reaches the CV value of single flapping wing (Fig. 5c). When the wings are very near to each other, they act as a single system and the added mass effect which depends on the shape of the body and the acceleration of the fluid is prominent. As the separation between the wings increases, the added mass effect decreases and consequently the initial peak in CV decreases (Figs. 5a and 5b). Fig. 5c shows that the CV of fore- and hindwings asymptotically reach CV of the single wing at large enough inter-wing distances. In the downstroke of the wings, the maximum influence of the delayed stall mechanism on LEV generation occurs between t/T = 0 and t/T = 0.5. As the inter-wing distance increases, the effect of forewing downwash on the LEV generation of the hindwing decreases. Consequently, LEV of the hindwing grows in size and the vertical force generated by the hindwing increases. For l < 1.3c, the width of the wake increases as the increase in l. For l > 1.3c, the width of the wake decreases as the increase in inter-wing distance. The decrease in the width of the wake reduces the force generation. It is clear from the Fig. 6 that LCS reveals structures which are otherwise hidden in vorticity plots.

313

Fig. 8 (a) and (b) Time varying CV of forewing and hind-wing respectively; (c) effect of interwing distance on cycle averaged vertical force coefficient.

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Fig. 9 Evolution of vorticity with time and LCS for l = 1.1c. Vorticity and LCS are plotted in alternating row.

of the forewing (Fig. 9b). This corresponds to the minima in the CV of the forewing at t/T ≈ 0.3. As the forewing moves away from the hindwing (Fig. 9c), the pressure on the lower surface of the wing increases and consequently CV increases (Fig. 9c) and reaches a local maxima at t/T = 0.3. At t/T = 0.3, the TEV of the forewing and CCW vorticity of the hindwing merge together, and form a region of low pressure between the two wings (Fig. 9c). As the hindwing starts its downstroke, it interacts with the merged CCW vortex, this wake capture increases the vertical force generation of the hindwing. In Fig. 9c, it is clear that the CV maxima at t/T = 0.6 is greater in magnitude than that of a single wing flapping due to the interaction with the merged vortex (Fig. 9d). After the wake capture, as the hindwing continues its downstroke, the downwash of the forewing and the LEV shed by the forewing result in a sudden fall in the vertical force. For the rest of its downstroke, the hindwing operates in a low pressure region created by the shed LEV of the forewing, the merged vortex and the growing LEV. This results in the sudden drop in the vertical force generation of the hindwing for t/T > 0.7. In comparison to a single wing flapping where the delayed stall mechanism generates much of the vertical force. Besides, a part of the shed LEV of the forewing and the CW shear layers merge with the LEV of the hindwing (Fig. 9d) and enhance the delayed stall effect. This results in a further reduction of pressure around the wing,

which is the major reason for the decrease in the vertical force generation of the hindwing. A pair of counterrotating vortices is shed in every flapping cycle. The vortex pair has low swirl component. This wake with predominant vertical velocity increases the stability of the body in hovering. Fig. 10 presents the vorticity contours at different interwing distances. The CV of the forewing slightly

(a) l = 1.1c

(b) l = 1.2c

(c) l = 1.3c

(d) l = 1.5c

Fig. 10 Effect of interwing distance on vortex structures: Vorticity contours for different l at the end of the downstroke. Strength of LEV and TEV increases with increasing inter-wing distance.

Srinidhi and Vengadesan: Lagrangian Coherent Structures in Tandem Flapping Wing Hovering

decreases initially, for l < 1.5c and remains nearly constant at greater distances. The effect is apparent as the vortical structures of the forewing look similar. The CV of the hindwing increases slowly for l < 1.5c and increases rapidly for l > 1.5c. For l ≥ 1.7c, there is no formation of the merged vortex, which reduces the CV peak of the hindwing at t/T = 0.6 (Fig. 8b) as the effect of wake capture is diminished. For l > 1.5c, due to the reduced wing-wing interactions, and decreased effect of merged vortex in reducing the pressure around the hindwing, the delayed stall mechanism becomes more effective in the force generation (Fig. 8b).

6 Conclusion We have investigated the effect of inter-wing distance and phase difference on the force generation of tandem flapping wings hovering in a quiescent fluid. Vorticity dynamics governing the physics is analyzed by visualizing vorticity contours and backward FTLE ridges. Wing-wing and wing-vortex interactions are studied. LCS helps in visualizing structures which are otherwise hidden in vorticity contours. In counterstroking, for most of its downstroke hindwing operates in the low pressure region created by the CW LEV shed by the forewing, as a result there is a drastic reduction in the generated forces. In in-phase and counter-stroking, increasing the inter-wing distance decreases the vertical force generated by the forewing. In-phase stroking generates maximum vertical force, but creates a wake with an appreciable horizontal velocity (swirl)[19] which gives rise to thrust. This shows in-phase stroking is best suited for forward flight of an insect where thrust is of paramount importance. It is also equally important during take-off when maximum vertical force is required. In counter-stroking forewing and hindwing shed vortices alternatively at the end of their respective downstrokes, which reduces the body oscillations facilitating stable hovering. It appears that counter-stroking with minimum inter-wing distance is the appropriate kinematics for hovering.

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