1 Introduction

An understanding of the aerodynamic characteristics of flapping wings plays an important role in designing an insect-like micro aerial vehicle (MAV). Insect flight is a recurring topic of study. Over the last decades, studies of aerodynamic and flapping mechanisms of flapping flyers have yielded remarkable results. However, these topics remain something of a mystery even though two-thirds of the world’s species use flapping flight (Shyy et al. 2008).

The leading edge vortex (LEV) is proposed as the reason for the aerodynamic force in insect flapping. The concept of a LEV contributing to a high lift force was first proposed by Maxworthy (Weis-Fogh 1973). Using mechanical flapping models, the LEV has been identified as a crucial source of lift for insects (Birch and Dickinson 2001; Ellington et al. 1996). The results of various experimental studies have proposed three mechanisms to explain aerodynamic performance in insect flight: delayed stall, rotational circulation, and wake capture (Dickinson et al. 1999; Sane and Dickinson 2001). A stable LEV is important to the generation and maintenance of high lift during flapping flight of insects (Birch and Dickinson 2001). At high Reynolds numbers, the LEV is stably attached to a wing during the downstroke and has a prominent helical vortex (Ellington et al. 1996). By constructing a dynamically scaled-up robot, researchers have quantified the LEV structure on an insect wing, such as that of a fruit fly (Birch and Dickinson 2001; Birch et al. 2004; Poelma et al. 2006), dragonfly (Lehmann 2009), and hawkmoth (Lua et al. 2010). Although the LEV was found in most insects, Ansari et al. (2009) showed that its characteristics seemed to differ for rotating plates according to Reynolds number. On the other hand, most previous works have been conducted to investigate the flow structure of a flapping wing model in the horizontal planes and for a symmetric stroke. Most recently, Park and Choi (2012) developed a dragonfly-like flapping wing that moved in an inclined plane. They showed that with a decreased rotational rate, the formation of a LEV through the wing translation was delayed and the size of the LEV decreased. However, the asymmetric strokes were generated because of the inline motion of the wing. The angular rotation remained symmetrical during the upstroke and downstroke; the model was unable to match the motion of a real insect wing.

It has been reported that the ground could contribute to the considerable advantages in aerodynamic performance during takeoff and landing (Rayner 1991). The ground effect was defined as when an animal or aircraft flies close above a plane surface and the aerodynamic properties of its wing are altered because of the interactions of the vortices with the wing and the wake with the surface (Rayner 1991). Previous studies on the ground effect have investigated the aerodynamic behavior of airfoils close to the ground by numerical simulation (Ahmed et al. 2007; Zhang and Molina 2011). For experimental study, Ahmed and Sharma (2005) studied the aerodynamic characteristics of a symmetrical airfoil at different ground clearances in a low-speed wind tunnel. They used a multi-tube manometer on the airfoil to measure the pressure and demonstrated that a high lift coefficient was obtained when the airfoil was close to the ground. However, there is a lack of in-depth studies on the flow around a moving object near the ground.

Few studies have been conducted regarding the ground effect of insect flying. Only a 2D numerical simulation has been reported in the literature. To deal with the ground effect for hovering motion, Gao et al. studied the unsteady forces and flow structures of an elliptic foil with oscillating translation and rotation near the ground (Gao et al. 2008). They showed that in the force enhancement regime, the interaction between the persistent vortex on the ground and the foil could enhance the LEV, and in the force reduction regime, the TEV moved away in the horizontal-upward direction because of the ground effect. However, the kinematic wing in their model had a generic sinusoidal motion and did not produce the kinematic motion of a real dragonfly wing. On the other hand, the wing kinematics in hovering flight were applied; it was not in the initiation flight (Gao et al. 2008). The flapping frequency, flapping angle, and angle of attack differ completely in hovering and initiation flight (Card and Dickinson 2008). Moreover, the distance between the wing center and the surface was randomly estimated in their model (Gao et al. 2008), and it has never been studied in the takeoff flight of a real dragonfly. Thus, there have been no studies of the ground effect for any real insect.

Therefore, the unsteady flow field of an insect wing and the relation to the ground effect remain to be studied. Coleopteran family insects are beetles that have large bodies and two pairs of wings: elytron (outer wing, fore wing) and hind wings (inner wing).The Rhinoceros Beetle (Trypoxylusdichotomus), which is commonly found in Japan and Korea, has been studied for its aerodynamic performance and three-dimensional wing kinematics (Le et al. 2010; Van Truong et al. 2012a, b). In a previous study (Van Truong et al. 2011), we found that the Rhinoceros Beetle takes off without jumping, which is uncommon for other insects. To completely characterize the fluid dynamic mechanisms involved in beetle flight as well as insect flight, it is necessary to investigate the entire vortex structure of beetle wings during takeoff. Thus, in this work we built a scaled-up electromechanical model of the flapping wing and investigated fluid flow around the beetle’s wing model. The present dynamically scaled mechanical model has a wing kinematics pattern that matches a real beetle’s wing kinematics during takeoff. In addition, we could systematically change the three-dimensional inclined motion of the flapping model through each stroke. We used digital particle image velocimetry (DPIV) with high spatial resolution and qualitatively and quantitatively studied the flow field around the wing at a Reynolds number of around 10,000.

2 Experimental apparatus and procedure

2.1 Development of wing kinematics and control

We designed and built a dynamically scaled robotic wing that can replicate the beetle’s wing motion. In this study we did not integrate the elytron into the robotic wing model, because the elytron’s force contribution was small compared to that of the hind wing (Sitorus et al. 2010). However, the elytron contribution was small only for hovering flight or very slow forward flight. During forward flight, the elytron-hind wing interaction enhanced the lift force and could not be neglected (Le et al. 2013). More importantly, since the elytron was positioned above the hind wing during takeoff, movement of the hind wing dominated the flow below the elytron. Thus, the ground effect did not impact the flow field around the elytron. Therefore, this work only investigated the flow around the hind wing. Basically, the kinematic parameters of the flapping hind wing are defined as the flapping angle, angle of attack, deviation angle, and stroke plane angle. Unlike other insects with figure eight or oval wing trajectories, such as the fruit fly (Sane and Dickinson 2001) and hawkmoth (Willmott and Ellington 1997), the wing tip path of the beetle’s hind wing is narrow and unusual. Therefore, it is difficult to mechanically replicate the wing trajectory of a real beetle’s hind wing for a flapping wing model. Additionally, in previous studies we showed that the deviation of the hind wing is small. The effects of small deviation angles on aerodynamic forces are minimal (Dickinson et al. 1999). Hence, the present study considers the flapping angle, geometry of the angle of attack, and stroke plane angle. We used 3D high-speed video techniques to quantitatively analyze the wings and body kinematics during the initiation periods of beetle flight. We captured and analyzed kinematics of the first four strokes. During takeoff, the height of the beetle from the ground (h/R) increased and the stroke plane angle decreased (Van Truong et al. 2011).

The horizontal and vertical velocities of the beetle were small. The dimensionless velocity motion expressed by the advance ratio J [J is the velocity of motion divided by the mean wing-tip speed (Ellington 1984)] was also small (from 0 to 0.03). Thus, the velocity of the flapping model was not considered in this study. Since the pattern of the angle of attack was nearly similar through four strokes, we superposed the kinematic data of the hind wings for four strokes. The flapping angles were averaged for the left wing and right wing. To employ the kinematic pattern close to a real beetle wing during takeoff, we fixed the flapping angle and the angle of attack for four strokes. We varied the stroke plane angle of the robotic wing for each stroke in accordance with a real beetle wing. We also changed the distance between the wing model and the ground for each stroke to match the change in the beetle’s vertical height during takeoff. Only the vertical height was studied in this investigation.

Figure 1a shows the schematic of the electromechanical system. We designed a flapping robotic apparatus that is driven by two stepper motors via the timing belts and pulley. The gear box (52 mm × 32 mm × 32 mm) contains a bevel-gear wrist, which was operated through the coaxial shaft. The wing model was attached to the flapper gear box by a wing holder and immersed in a tank (90 cm width × 60 cm height × 60 cm length). The shafts are long enough so that the model wing is close to the bottom of the tank. The stepper motors were connected to the motor controller. The step angle of the motors was set to 0.018°. The motion of the stepper motors was controlled by self-written software. The distance between the robotic wing and ground was created by changing the height of the support bars (values h 1 and h 2). Figure 1b defines the geometry of the angle of the mechanical fly wing. The origin of the coordinate system is located at the center of the shaft. The stroke plane angle is the angle between the horizontal plane and the plane formed by three points: the wing base and the wing tip at the end of both the downstroke and upstroke. The flapping angle or sweeping angle φ is defined as the angle between the line that joins the wing base and wing tip measured at the ends of the downstroke and upstroke. The angle of attack is the geometric angle that the wing chord makes with the stroke plane. The stroke plane angle β of each stroke was set by varying the values of \(h_{1}\) and \(h_{2}\):

Fig. 1
figure 1

a Diagram of the robotic beetle set up. The dynamically scaled robotic wing consists of a flapping wing driven by two stepper motors. The wing was immersed in a clear glass tank of glycerin/water mixture. b Definitions of the wing kinematic parameters of the wing robofly

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$$\beta = ac\tan \left( {\frac{{h_{1} - h_{2} }}{d}} \right)$$
(1)

To obtain an accurate scale of the real beetle wings during takeoff, the distance between the wing and the ground should be maintained. Figure 2 shows a beetle at the beginning of the upstroke in takeoff and the model wing near the bottom of the tank, which functioned as the ground. The heights of the wing base (h) and wing length (R) in the real beetles were determined from the experiment in our previous study. The non-dimensionless ratio h/R in the real beetles was defined and implemented for the robotic wing. Table 1 presents the stroke plane angle and the ratio h/R for four strokes. During the beetle’s takeoff, the distance h and stroke plane angle β varied according to time. To use these parameters for the dynamically scaled-up robotic wing required two more degrees of freedom for the distance and the stroke plane angle. However, it is extremely difficult to construct the mechanical and electrical systems to accommodate varying the distance and the stroke plane. Due to the small advance ratio (J), the unsteady effect could be neglected, and the distance h and stroke plane angle could be averaged. h was determined as the distance of the wing base from the ground at the middle of the stroke. The stroke plane was defined as the plane formed by the wing base and the wing tip at the lowest and highest positions of the stroke cycle (Figs. 3, 4).

Fig. 2
figure 2

Diagram showing the geometry of hind wing near the ground in the real beetle (a) and the robofly (b). R is the wing radius; h is the distance between the wing root and ground

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Table 1 The geometry of hind wing near the ground in the real beetle shown in Fig. 2
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Fig. 3
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Photograph of experimental setup

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Fig. 4
figure 4

Schematic diagram of DPIV experimental setup

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The outer shaft rotates the gear box to control the flapping angle. The angle of attack is controlled by a bevel-gear driven by both the inner and outer shaft. To mimic the kinematic pattern of a real beetle hind wing, we programmed the motion of the stepper motors for many delay times and various speeds in one cycle. The angular movement of the model wing was calculated via the gear box ratios, pulley ratios, timing belt, and the speed of the stepper motors. In developing the stepper motors’ programming, we attached an angular ruler to the gear box and the outer shaft (Fig. 5). When the model was actuated, the actual rotation angle of the wing was examined using the angle markers. The programming code was adjusted step by step to achieve the kinematics of a real beetle hind wing. Finally, we conducted a high-speed videography experiment to quantitatively and qualitatively validate the programmed kinematics and the actual motion of the wing. Figure 5 presents a diagram of the wing kinematic experiment. Two high-speed video cameras (Photron, APX) were used to obtain the three-dimensional kinematics of the robotic wing. The images were recorded at 125 fps with a pixel resolution of 1,024 × 1,024. The high-speed cameras were connected to image processors and personal computers. Two cameras were positioned to capture the top side view of the model wing. Three 5-mm white dots of non-toxic paint were marked on the leading edge, the trailing edge at the middle span, and the tip of the model wing. A three-dimensional coordinate of each point was determined using the direct linear transformation method (DLT). The cameras were calibrated with a 64-point calibration frame (measuring 15 cm × 15 cm × 15 cm in x, y, and z coordinates). The coordinates of the digitized points were input to a Matlab script (Hedrick 2008), and the kinematic data were filtered with a Butterworth filter at a cutoff frequency nearly four times greater than the flapping frequency. This experiment was performed in air. To examine the load impacted to the model wing as it moved in fluid, we measured the angle of attack at the middle of the downstroke and upstroke. A high-speed camera and its lens were oriented perpendicular to the model wing at the middle of the downstroke and upstroke while the model wing was beating in the tank. The angle of attack at those positions was manually calculated using AutoCAD. There was no difference in the angle of attack when it flapped in the air or fluid. The flapping angle (φ) and the angle of attack (α) were approximated by the high order series function as follows:

Fig. 5
figure 5

Schematic diagram of wing kinematic experimental setup

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$$\begin{gathered} \varphi (t) = \sum\limits_{n = 0}^{n = 6} {\left[ {\varphi_{cn} \cos (nkt) + \varphi_{sn} \sin (nkt)} \right]} \hfill \\ \alpha (t) = \sum\limits_{n = 0}^{n = 6} {\left[ {\alpha_{cn} \cos (nkt) + \alpha_{sn} \sin (nkt)} \right]} \hfill \\ \end{gathered}$$
(2)

2.2 Model wing platform

The span b, the mean chord length c, and the aspect ratio (AR = b/c) of the real beetle’s hind wing are 45, 14.6 mm, and 3.08, respectively. The wing beat frequency varied for each stroke from 38 to 42 Hz. In this study, the mean wing beat frequency of the hind wings was chosen at 40 Hz for all strokes. The model wing was fabricated from transparent acrylic sheets and was cut in the shape of the beetle hind wing using AutoCAD and a precision CNC machine with a resolution of 0.001 mm (M300S CE, Woo Sung E&I Co., Ltd., Korea). The radius, the mean chord, and the thickness of the model wing were 180, 58.4, and 3.2 mm, respectively. Only a rigid wing was considered for this work. The radial distance of the wing tip to the flapping axis is 20 cm. The Reynolds number was calculated in this study using Eq. (3).

$$R_{e} = \frac{{U_{t} c}}{\upsilon } = \frac{2\theta nRc}{\upsilon }$$
(3)

where \(U_{t}\) is the mean wing tip translation velocity, c is the mean chord length, R is the wing length, θ is the wing beat amplitude, n is the stroke frequency, and ν is the kinematic viscosity of the fluid (Ellington 1984). The Reynolds number of a real beetle is 10,000.

2.3 DPIV measurement

To quantify the instantaneous flow structures and measure the flow field of the flapping wing, we used a custom continuous-laser DPIV system. Figure 3 shows a photograph of the components used and the arrangement of the experimental setup. The wing was transparent, and the laser could pass though the wing with the consistency of a laser sheet strength and direction. The optics consists of a convex mirror and concave mirror to transform the beam to a laser sheet. The fluid in the clear glass tank was seeded with fluorescent polystyrene particles (Molecular Probes FluoSpheres® polystyrene microspheres, 1 μm in mean diameter). To prevent particles from sinking and to generate uniform particles in the fluid, we used the glycerin/water mixture (volume fraction of 0.4954) as working fluid. The experiment was performed when the fluid was static without convective flow inside. The kinematic viscosity of the working fluid was measured using a viscometer (SV10-100 Sine-Wave VibroViscometers, A&D Korea Co., Ltd.). During the experiment, the temperature of the glycerin/water mixture was 22 °C and its viscosity (\(\upsilon\)) was 5.86 × 10−6 m2/s (Cheng 2008). Using Eq. (3) and the Reynolds number of the model wing, which is equivalent to that of a real beetle, the flapping frequency is given as 1 Hz. Images were acquired with a high-speed camera (Photron APX) fitted with a 50-mm Nikon lens. When the images were captured at a low frame rate, the particle was prolongated. To capture the accurate movement of particles and comprehend the motion of fluid around the wing, the images were recorded at 1,000 fps with a pixel resolution of 1,024 × 1,024.

The high-speed camera was placed perpendicular to the laser sheet as shown in Fig. 4a. The laser sheets were positioned at 0.25R, 0.5R, and 0.75R. In this work we analyzed the flow structures around the wing at four important periods of the wing beat cycle: the pronation period, supination period, and middle of downstroke and upstroke. The camera and laser were positioned so that the wing was in front of the camera. Figure 4b, c shows the arrangement of the camera and laser at the middle of the downstroke and upstroke, and in the supination period, respectively. Similar to the supination period, in the pronation period the camera was placed on the left side of the tank, and the laser was placed in front of the tank. To ensure the spanwise direction of the wing was perpendicular to the camera lens at the middle of the downstroke and upstroke, we set the initial position of the model wing (Fig. 4b, c). The spanwise direction was defined by the line from the root to the tip of the wing. We mounted an angular circle to the outer shaft to adjust the initial position of the model wing, as shown in Fig. 5b. To define the exact position of the model wing passed though the laser sheet (t = 0T, t = 0.25T and t = 0.75T, t = 1T), we manually measured the angle of attack of the model wing from the recorded images. The angle of attack differs at each time step through the stroke; the position of the model wing could be defined on that basis.

Two images were captured and then processed by commercial PIV software (DaVis 7.1, Lavision Inc.) using a standard cyclic FFT-base cross-correlation algorithm. To acquire magnified images we mounted a precision ruler at the center of and parallel to the thin laser sheet. The ruler was moved to the spanwise positions of 0.25, 0.5, and 0.75 R corresponding to the laser sheet position. For the cross-correlation procedure we used an initial interrogation window size of 32 × 32 px in the first pass and 16 × 16 px in the subsequent passes, followed by a pixel intensity peak with 50 % overlap. For a 0.9 m/s wing velocity at the middle span, the PIV results showed a value of 0.904 m/s, which produced an error of about 4 %. DPIV analysis was done using a written Matlab script and drawn using Tecplot 10. Three stroke cycles were recorded and averaged to calculate the flow velocities. The flow field data achieved showed good repeatability for each stroke cycle.

3 Results

3.1 Wing kinematics

Figure 6 shows the kinematics of the hind wing and model wing in one cycle. The shaded region corresponds to the upstroke period. The non-dimensional time (t/T) is 0–0.5 for the upstroke and 0.5–1 for the downstroke. The flapping angles of the hind wing and model wing were 170° and 171°, respectively. The variation of the stroke position angle with respect to time was approximated as a sine function. The geometric angle of attack (rotating angle) varied from 10° to 140°, which was wider than that in the free forward flight condition. Figure 6c shows the periodic motion of the hind wing during a beating cycle. The kinematics of the model wing was close to that of the hind wing of a real beetle. The mean rotating angle of the beetle hind wing and the model wing was 89.82° and 87.85°, respectively, which is a difference of 2.2 %. The mean flapping angle of the beetle hind wing and the model wing was 86.87° and 87.18°, respectively, which is a difference of 0.36 %.

Fig. 6
figure 6

The variation of flapping angle (a) and rotating angle (b) (geometric angle of attack) of the beetle hind wing and model wing. c The wing movement as two-dimensional projections

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3.2 Wake patterns

In order to visualize a visible course description of flow structures, we recorded the wing motion at a low frame rate (125 Hz). Figure 7 illustrates the wing-wake interaction produced during the pronation period in the second stroke. The laser sheet sliced perpendicular to the model wing at the middle of the span. The vortices were clearly visible in the fluid. A strongly accelerated structured wake is generated when the wing rotates. The black region at the bottom of the images represents the ground. To prevent the reflection of the laser from the bottom surface of the tank, we placed a black aluminum plate of 3 mm thickness at the bottom. The LEV and TEV are clearly visible at the end of the downstroke (t/T = 0.952). The TEV detached from the wing at the middle of the stroke reversal while a circulation was formed around the wing (t/T = 0). The LEV and TEV were formed at the beginning of the upstroke (t/T = 0.048), at which point the TEV left from the downstroke interval remained.

Fig. 7
figure 7

Time sequence of the wake produced by a model wing in ground effect. Images were taken using a high-speed camera captured at 125 Hz. The wing was rotating during pronation period. The red lines represent the visibility of chord wise wing. The red dots the leading edge. The black region at the bottom each image indicates the ground. The yellow pictograms display the location and spin of visible vortices in all images

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3.3 Flow structures

In order to provide a comprehension of the flow structure in the ground effect, we performed the experiments without the ground effect. We lifted the flapping wing system so that the model wing was at the middle of the tank. The distance from the wing tip to the wall was 20 cm (equal to the rotational radius). In comparison to previous studies (Ansari et al. 2009; Beem et al. 2012; David et al. 2012; Ishihara et al. 2009; Kim and Gharib 2010; Kurtulus et al. 2008; Liang et al. 2011; Lu and Shen 2008; Lu et al. 2006; Zhao et al. 2010), the walls of the tank were sufficiently wide to ensure the absence of edge effects. The stroke planes of the model wing were unchanged, and the same wing motion was repeated.

Figure 8 shows the evolution of flow structure during pronation of the first stroke at middle span (h/R = 0.5, β = 23°). The sequences started with the time t/T (t*) = 0.025. The vertical blue line separated the wing motion with and without the ground. The first column in the layout of images shows the vorticity, and the second column shows the streamline patterns of the non-ground condition. Meanwhile, the third and fourth columns show the reverse. When the wing was at the middle of the rotation periods (t* = 0.950), the LEV and TEV were obviously present. When the wing is rotated clockwise, the pronounced negative level of the vorticity in the LEV is reduced (t* = 0.975). By time t* = 0 (end of the downstroke), the circulation was generated around the wing. This circulation bounded the wing, producing rotational circulation at the stroke reversal (Lehmann 2004). As the wing entered the second upstroke, that circulation remained around the wing (t* = 0.025) and moved downward to the trailing edge. The vortex began to roll up to form a new LEV in the reversed stroke at t* = 0.05, and the size and strength of the LEV intensified gradually with time.

Fig. 8
figure 8

Contours of the instantaneous vorticity and corresponding streamline patterns around the middle span of model wing (0.5R) during the pronation period of the first stroke. The black region at the bottom of each image indicates the ground

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In order to examine detailed flow characteristics, the streamline of each image is plotted. The streamline patterns provide useful information about the instantaneous flow field near the wing (Lua et al. 2008). The center of the streamline patterns was located very near the leading edge at t* = 0.950. It then scrolled down along the chord at 0.25 chord (t* = 0.025), 0.75 chord (t* = 0.050), and the trailing edge (t* = 0.0750). The movement of the circulation from the leading edge to trailing edge could reduce the rotational lift force when the angular velocity was reduced (Sane and Dickinson 2002; Sun and Tang 2002). Consequently, the movement of that circulation emanated from the TEV at the next upstroke (t* = 0.050). The TEV moved further downward because of the downwash flow, and was stronger and larger (t* = 0.075). This appeared to be aided by the induced velocity from the new TEV. At this time there were three different vortices around the wing: the positive LEV, negative TEV, and positive TEV_1. Plots of vorticity and streamlines provide a quantitative view of flow dynamics around the model wing in ground and non-ground conditions.

The evidence of the dissimilarity of non-ground and ground conditions is shown at the time t* = 0.950. At t* = 0.950, the LEV on the wing in the ground condition is larger than that in the non-ground condition. The LEV contributed to a high lift force (Birch and Dickinson 2001). The lift contribution in the ground case could thus be greater than that in the non-ground case. On the other hand, corresponding to the streamline patterns, there are two TEVs in the ground condition at this instant. Two TEVs were also present at the beginning of the upstroke (t* = 0.05). Meanwhile, there is only a single TEV in the non-ground case. At t* = 0.975 the LEV vortices in the non-ground case are more slender than that of the ground condition. Additionally, there was more bound circulation for the ground case at t* = 0.0. Due to the flow intact with the ground, the flow rolled up into the larger TEV in comparison to the non-ground case. At t* = 0.075, the TEV left from the previous stroke was shedding downwash in the non-ground case, but it remained in the ground case.

Figure 9 depicts the evolution of flow structures in the second stroke (h/R = 0.6, β = 20°). The vortex structures in the non-ground case are similar to that of the first stroke. In terms of the flow structure with the ground case, the patterns were similar to the non-ground case. There is a little disturbance of the TEV remaining from the previous stroke when the wing was moving up (t* = 0.025, t* = 0.50, and t* = 0.075). The size of the LEV is the same as that of the non-ground case. It was also found that the vortex structures during the wing rotation in non-ground cases during the third stroke (β = 13°) (Fig. 10) were consistent with those in the first and second strokes. During the downstroke, vortex development is the same between the ground and non-ground cases (t* = 0.950, 0.9750). However, the LEV changed as it traveled toward the end of the downstroke (t* = 0). There are two vorticity regions: a small region layer with a negative vortex located above the leading edge and a big clock-wise circulation at 0.25 chord. According to the streamline patterns, both vortices were covered by the outside vortex. The small vortex (LEV_1) was generated because of the separation of the LEV during the vortex roll-up process. When the wing reversed its direction to the upstroke (t* = 0.025), that vortex grew gradually. As a consequence, at t* = 0.050 there are four vortices: the negative-sign vorticity (LEV_1) behind the wing, the positive LEV, the negative TEV, and the positive TEV left from the downstroke. The flow structures of ground and non-ground cases in the fourth stroke (β = 4°, h/R = 0.72) during the pronation period are similar to the third stroke of the non-ground cases, which we do not present here. Hence, we can conclude that at h/R = 0.72 the wing was not influenced by the ground effect.

Fig. 9
figure 9

Contours of the instantaneous vorticity and corresponding streamline patterns around the middle span of the model wing (0.5R) during the pronation period of the second stroke. The black region at the bottom of each image indicates the ground

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Fig. 10
figure 10

Contours of the instantaneous vorticity and corresponding streamline patterns around the middle span of the model wing (0.5R) during the pronation period of the third stroke. The black region at the bottom of each image indicates the ground

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Figure 11 provides an overview of the flow structures of the middle spanwise for four strokes at the middle of the upstroke. The angle of attack at the middle of the upstroke is 43°. The stroke planes of the first, second, third, and fourth strokes are 23°, 20°, 13°, and 4°, and the angle of the chord with the horizontal plane (\(\alpha_{m}\)) is 66°, 63°, 56°, and 47°, respectively. A strong LEV is formed on the upper wing. The LEV also remains without shedding. The streamline patterns indicate the stability of the LEV at high angles of attack and the similarity of the LEV structure in the four strokes (non-ground case). The TEV is generated and moves downstream. The effect of the ground effect can be seen at the first and second strokes. In the first stroke, the magnitude and size of the LEV in the ground condition are larger than those of non-ground cases. Due to the great distance between the wing and ground, there is little influence on the flow structures of the wing. It is expressed through the third and fourth strokes: the size of the LEV is similar in non-ground and ground cases. Most notably, there are two positive vorticity regions attached on the surface of the wing in the third stroke when the ground is present. This vortex was named dual LEV, and it was revealed on a flapping wing model in a previous study (Lu et al. 2006). We did not observe the dual LEV on the model wing without the presence of the ground.

Fig. 11
figure 11

Contours of the instantaneous vorticity and corresponding streamline patterns around the middle span of model wing (0.5R) at the middle of the upstroke of four strokes. The black region at the bottom of each image indicates the ground

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Figure 12 emphasizes the structure of a dual LEV on the middle span of the hind wing at the middle of the upstroke in the second stroke. Unlike the non-ground case, the LEV area was no longer connected and was separated into sub-regions. The right images quantify the derivation of a course description for the flow structure at that moment. The vortex reconstruction of the wake highlights the strong interaction of the flow with the ground. The flow visualization is consistent with the PIV results. The form of the streamline patterns is similar to the swirl of the vortex in the flow visualization. The experiment by Lua et al. (2008) found that the dual LEV can be created when the angle of attack is larger than 30°. However, the generation of dual LEV was not found in the four strokes of the non-ground case, where the angle of attack was 43°. It was also not presented in the first, third, and fourth strokes of the ground case. The LEV structure is similar in the third and fourth strokes with and without the ground.

Fig. 12
figure 12

Analysis of wake capture of the middle span of model wing (0.5R) at the middle upstroke of the second stokes. The red lines represent the visible of chordwise wing. The red dots indicate the leading edge. The black region at the bottom of each image indicates the ground

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Figure 11 also shows the effect of the angle \(\alpha_{m}\). When \(\alpha_{m}\) is increased, the TEV sheds the trailing edge far away. In the non-ground case, the LEV was largest at \(\alpha_{m}\) = 56 (stroke3), and was small when \(\alpha_{m}\) increased or decreased (strokes 2, 3, and 4). Figure 13 further illustrates the effect of \(\alpha_{m}\) and indicates the contours of the instantaneous vorticity around the middle span of the model wing (0.5R) at the middle of the downstroke for four strokes. It is seen that the LEV is a function of \(\alpha_{m}\). In the first and second strokes the LEV is small, and it grows as \(\alpha_{m}\) increases. Regarding the flow structures of the ground case, there is no difference from the non-ground case at this instant. However, the LEV at the first and second strokes is much larger than that of the non-ground case. The LEV is the primary lift force for a flapping wing (Birch and Dickinson 2001). Due to the effect of \(\alpha_{m}\), the LEV in the upstroke is stronger and larger than in the downstroke (see Figs. 11, 13).

Fig. 13
figure 13

Contours of the instantaneous vorticity around the middle span of the model wing (0.5R) at the middle of the downstroke of four strokes. The black region at the bottom of each image indicates the ground

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Figure 14 shows the wake velocity in the middle of the downstroke and upstroke for four strokes of the model wing. The small upper region of the leading edge corresponds to a counter-clockwise LEV. During upstroke with high angle \(\alpha_{m}\), the velocity of the fluid above the wing rearward was higher than that of the downstroke. This figure also shows an induced downward flow. The fluid on the ground was unperturbed during the upstroke, while it was affected during the downstroke. This suggests that the LEV might be affected during the downstroke.

Fig. 14
figure 14

Analysis of flow field velocity of the middle span of the model wing (0.5R) at the middle of the downstroke and upstroke. The black region at the bottom of each image indicates the ground. The direction of wing translation is from right to left for the downstroke and left to right for the upstroke, respectively

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3.4 Flow variations with span

Figure 15 shows the spanwise flow for three cross sections (0.25R, 0.75R, and 0.5R) along the span of the model wing in supination and pronation periods of the first stroke. The flow pattern in the supination period of the second, third, and fourth strokes, and all the non-ground strokes are similar to the first stroke. We could not find any effects of the ground on the model wing in the supination period. Corresponding to an increase in flapping velocity, the leading and trailing vortices grow progressively in moving outboard spanwise position. The TEV was shedding near the tip of the model wing. The LEV formed from the root, developed more along the spanwise, and was well attached at the end of the upstroke (t* = 0.475, t* = 0). As the wing traveled into the downstroke, the LEV detached near the wing tip, but remained attached near the root (t* = 0.525). The LEV and TEV are almost detached from the wing and turn to lie below the wing. This phenomenon is called wake capture, in which the LEV and TEV are detached and interact with the wing after turning (Dickinson et al. 1999). After the wing reversed direction, there was a significant difference in the flow field between various spanwise positions (t* = 0.550). The negative LEV formed at 0.75R, while eventually there were only wake captures at 0.5R and 0.25R. The LEV gives a stable attachment from the root to the tip at the end of the upstroke (t* = 0.475) and downstroke (t* = 0.975). Due to the tilting of the wing following the stroke plane, the gap between the wing and ground was reduced from root to tip in the pronation period. As a result, the TEV decreased in size at greater radial distances from the root. At the beginning of the upstroke, the LEV was generated and enlarged from near the middle to the tip of the wing (t* = 0.050) but was not formed near the root.

Fig. 15
figure 15

Contours of the instantaneous vorticity along the span model wing during the supination period (first column) and pronation period (second column) of the first stroke. The black region at the bottom of each image indicates the ground

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The stable spanwise attachment of the LEV is evident at the middle of the upstroke (Fig. 16) and downstroke (Fig. 17). Figure 16 shows overviews of a spanwise structure with the hind wing at the middle of the upstroke. The figure shows that \(\alpha_{m}\) influences the spanwise vortex structure. In the first and second stroke \(\alpha_{m}\) is small, and the growth of the LEV in the spanwise direction is smaller than that for the third and fourth strokes, which have large \(\alpha_{m}\). The LEV expands significantly in the upstroke, where \(\alpha_{m}\) is much greater than that of the downstroke. The features of the LEV at increasing radial distances have a conical form. In particular, the dual vortices were enlarged during the second stroke along the direction of the outboard, and the main LEV was still attached to the wing. Near the root (0.25R) there was only a single LEV.

Fig. 16
figure 16

Contours of the instantaneous vorticity along the span model wing at the middle upstroke of four strokes. The black region at the bottom of each image indicates the ground

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Fig. 17
figure 17

Contours of the instantaneous vorticity along the span model wing at the middle downstroke of four strokes. The black region at the bottom of each image indicates the ground

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4 Discussion

4.1 Flow structure in a physical beetle model

Using a dynamically scaled mechanical model of a flapping wing, we quantified the flow structure of the beetle’s hind wing during takeoff. The kinematics of our beetle’s wing model were similar to those of a real beetle’s hind wings. The beetle hind wing motion was conducted at Re = 10,000. The results for general beetle wing kinematics show an asymmetric pattern of angles of attack. Furthermore, the angle of attack of the flapping model was not constant during the translation periods.

In this study we sought to understand the flow patterns around the beetle’s hind wing for an initial flight condition. A series of experiments were conducted to investigate the effects of distance and wing kinematics on the aerodynamics of a beetle’s hind wing. The major flow structures of a beetle hind wing are similar to the major vortical structures found in the common unsteady mechanism of aerodynamics of other insects: wake capture, delay stall, and rotational circulation.

In general, the phenomenon in which the LEV and TEV persist after a stroke reversal is called wake capture. The wing-wake is proposed to contribute significantly to the aerodynamic forces after stroke reversal (Dickinson et al. 1999). Consistent with experiments, the CFD simulation (Ramamurti and Sandberg 2002; Sudhakar and Vengadesan 2010) has presented evident wake capture in an insect flight model. This mechanism has been discovered on a dynamically scaled mechanical model of a flapping wing (Dickinson et al. 1999). In real insect flight, it has been demonstrated on free-flying butterflies (Srygley and Thomas 2002). In beetles the LEV and TEV shed into the wake placed below the hind wing (Fig. 15). The wake shed from the previous stroke enhances the velocity and acceleration fields, resulting in higher aerodynamic forces following stroke reversal (Sane 2003). The wake capture or wing-wake interaction occurred in both supination and pronation periods (Dickinson et al. 1999). However, in the takeoff flight of the beetle, this phenomenon only occurred in the supination periods (as shown in Fig. 15). It is explained by the asymmetric flapping motion of the hind wing. The wake-capture concept for fight force enhancement occurred in hovering fight with low advance ratios (Lehmann 2004). During forward flight, the wake was stretched out in space, so the wake capture was negligible (Lehmann 2004). During takeoff the advance ratios of the beetle are small.

Rotational circulation occurred when the hind wing rotated in the pronation periods, as shown in Figs. 8, 9, and 10. This phenomenon has been observed on a robotic Drosophila (Dickinson et al. 1999). Circulation bound the wing and traveled from the leading to trailing edges with an increase of their angular velocity; as a result, the rotational force increased (Sane and Dickinson 2002). The benefit of rotational circulation (which is similar to the Magus effect) is remarkable for flying insects (Lehmann 2004). Unlike previous studies that showed rotational circulation occurring in stroke reversal, the beetle hind wing showed rotational circulation in the pronation period only.

Delay stall (which is the most important of all unsteady mechanisms identified for flapping wings) was used to produce large lift at a high angle of attack. It has been experimentally studied in dynamically scaled flapping wings (Dickinson et al. 1999) and real free-flying insects (Thomas et al. 2004). The delay stall contributed 65 % of the total weight support of the hawkmoth (Bomphrey et al. 2005) and 88 % of the total lift force to fruit flies (Wu and Sun 2004). The presence of the LEV was clearly observed in the four strokes of the beetle hind wing during takeoff. The delay stall is remarkable during upstroke (Fig. 17) and the rotation period (Figs. 8, 9, 10), in which the angle of attack is large. The LEV formed at the beginning of strokes and grew in size (Figs. 7, 8, 9). The TEV detached and shed into the wake (Figs. 8, 9, 10, 11).

Furthermore, the important feature of the vortex structure on the beetle’s hind wing is the stability of the LEV. The LEV was formed at the beginning of the stroke and persisted until the end of the downstroke (Figs. 8, 9, 10) and upstroke (Fig. 15). The persistence of the stable LEV was observed along the span. In general, the LEV becomes gradually stronger moving outward along the wing.

4.2 Effect of ground

In this experimental study we provided an emerging picture of the flow dynamics of the beetle’s hind wing with and without the ground effect. To evaluate the ground effect, we also performed experiments in the absence of ground. In general, the hind wing has the basic flow structure of a physical insect model when it flaps near the ground. With the ground present, when the wing is nearest to the ground (stroke1) a pair of TEVs is generated at the end of the downstroke and beginning of the upstroke. In the third stroke, as the wing enters into the upstroke, the flow is drawn into two vortices: a vortex left above the wing and circulation bounding the wing (Fig. 10). In particular, the dual LEV on the wing was revealed at the middle of the upstroke in the third stroke. This observation contradicts the previous study, as Lua et al. (2008) found that the dual LEV was present at high angles of attack. Nevertheless, the LEV behaves differently at small scales such as that of flying insects or at various Reynolds numbers (Ansari et al. 2009). In contrast to the work by Lua et al., the dual LEV was not found in the other strokes and without ground in the beetle flight model during takeoff, which also has a high angle of attack. In this case, the dual LEV forms because of the optimal distance between the ground and wing. More importantly, the LEV is large at the middle of the downstroke in the first and second strokes in comparison to the non-ground case. With a greater focus on the strength of the LEV, as the wing travels to the end of the downstroke, the hind wing generates a stronger and larger LEV than that of the non-ground case. It could be concluded that the LEV was enhanced during half the downstroke of the first and second stroke. Hence, the vertical force could be enhanced to lift the beetle in takeoff. Note that the hind wing beat with an inclined stroke. In the supination periods, the distance between the wing and ground is large enough that the flow around the wing is not impacted by the ground. As the above results show, in the supination period the flow structure around the wing is similar to the non-ground case. The gap between the wing and ground is reduced as the wing moves downward. One explanation for the larger LEV is that during this time the fluid between the wing and ground is pressed, and the fluid then receives an impulse from the ground and is transported upward to the leading edge. Therefore, the LEV increases in size. The results also show that in the fourth stroke (h/R = 0/72) there is no difference in the vortex structure between the ground and non-ground. This observation could help researchers figure out the hovering flight condition when they conduct experiments on a dynamically scaled flapping wing or carry out simulation work.

5 Conclusion

In this work we have presented a dynamically scaled beetle hind wing model that replicates the general stroke pattern of beetle hind wing kinematics during takeoff flight. Studying the aerodynamic mechanism of the beetle in initial flight, we found that the main difference from other insects could be summarized by the rotational circulation in pronation periods and the wake capture in supination periods. Furthermore, the evolutions of flow structures around the hind wing and the physical constraint underlying ground-wake interaction have been explained. The LEV was enhanced during the half downstroke because of the ground effect, and the lift force could thus be increased to lift the beetle during takeoff. Although this is a two-dimensional study, the present studies may broaden our understanding of the aerodynamics and flow structures of insects in takeoff flight.