1 Introduction

Lift-generation outcome of wing clapping has been substantiated both experimentally [1] and theoretically [2]. The mechanism has been used in laboratory MAVs to increase the lift- and thrust-generation efficiency (see, for example, [3]). Computational analysis of flapping-wing aerodynamics with wing clapping requires the accuracy of interface-tracking (moving-mesh) methods, flexibility of being able to deal with the topology change (TC) in the fluid mechanics mesh when the wings are brought into contact, and the computational practicality of accomplishing these in 3D analysis. The space–time (ST) interface-tracking method with TC (ST-TC), introduced in [4], satisfies these requirements. The ST-TC method is a new version of the deforming-spatial-domain/stabilized ST (DSD/SST) method. The DSD/SST method was introduced in [5–7] as a core computational technology for flow problems with moving boundaries and interfaces and since then has been gaining more and more power [8–12]. Its moving-mesh feature makes it comparable to the Arbitrary Lagrangian–Eulerian (ALE) formulation [13], which is the most widely used moving-mesh method, with many successful applications in fluid–structure interaction (FSI) (see, for example, [12, 14–52]).

Computations based on the DSD/SST method also have been very successful in a number of classes of fluid mechanics and FSI problems, and formidable computational challenges were addressed with the special methods targeting those classes of problems. Examples, with the cited references reporting recent computations, are spacecraft aerodynamics [53], spacecraft parachutes [12, 54–60], cardiovascular fluid mechanics [4, 60–67], flapping-wing aerodynamics [4, 60, 64, 68–70], wind-turbine aerodynamics [60, 64, 71, 72], and data compression [73].

Both the ALE and DSD/SST methods possess the desirable features of moving-mesh methods, including mass conservation across the interface and better resolution of the boundary layers. The boundary layers are resolved more accurately because as the fluid–solid interface moves, the refined-mesh region follows the interface. These desirable features do not come easily or do not come at all with the interface-capturing (nonmoving-mesh) methods. Comments on and examples for what the DSD/SST method brings to the table beyond what the ALE method does can be found in [4], and also in [4, 10–12, 60, 64, 66, 68–73], including examples from aerodynamics of flapping wings and wind turbines and fluid mechanics of heart valves.

In its ST-TC version, the DSD/SST method is enhanced with a master–slave system that maintains the connectivity of the “parent” fluid mechanics mesh when there is contact between the moving interfaces. With that enhancement and because of its ST nature, the ST-TC method can deal with an actual contact between solid surfaces in flow problems with moving interfaces. It accomplishes that while still possessing the desirable features of moving-mesh methods, the key desirable feature being better resolution of the boundary layers.

Even before its ST-TC version, the DSD/SST method, with effective mesh update, was already able to handle moving-interface problems when the solid surfaces are in near contact or create near TC. Flapping-wing aerodynamics of an actual locust [12, 60, 64, 68], with the forewings and hindwings crossing each other very close and creating near TC, is an example of successfully computed problems. Flapping-wing aerodynamics of a micro aerial vehicle (MAV) with the wings of an actual locust [60, 64, 69, 70] is another example. However, as commented in [4], in some moving-interface problems with contact between the solid surfaces, the “nearness” that can be modeled with a moving-mesh method without actually bringing the surfaces into contact might not be “near” enough for the purpose of solving the problem. It was mentioned in [4] that fluid–solid interface-tracking/interface-capturing technique (FSITICT) [74] was motivated by such FSI problems.

In the FSITICT, we track the interfaces wherever and whenever we can with a moving mesh, and capture over that moving mesh the interfaces we cannot track, specifically the interfaces where and when we need to have an actual contact between the solid surfaces. As commented in [66], essentially, the FSITICT is based on giving up on the interface-tracking accuracy in the parts of the domain where and when we expect an actual contact. As also commented in [66], while this is better than giving up on the interface-tracking accuracy everywhere in the domain by using purely an interface-capturing method, the flow would not be represented accurately between the solid surfaces as they close.

The ST-TC method does not give up on interface-tracking accuracy even where there is an actual contact between solid surfaces or other TC. It does not require unstructured ST mesh generation. Details of the ST-TC method can be found in [4], together with conceptual examples and 2D test computations with models representative of the classes of problems targeted with the method. In [66], the ST-TC method was extended to 3D fluid mechanics computation of an aortic valve with coronary arteries and a mechanical aortic valve. Here we show how the ST-TC method enables 3D computational analysis of flapping-wing aerodynamics of an MAV with wing clapping. We use a model dragonfly MAV as the test problem. In the analysis, the wings are brought into an actual contact when they clap. The computational analysis is presented in Sect. 2, and the concluding remarks are given Sect. 3.

2 Dragonfly MAV

2.1 Geometry and flapping-motion modeling

The design of the wings is similar to the design in a toy MAV [75], and the body is the same as the MAV body in [69]. The span of the single wing is \(46.7~\mathrm {mm}\), and the minimum, maximum and average chord lengths are \(16.2\), \(19.2\) and \(17.6~\mathrm {mm}\), respectively (see Fig. 1). The wings have zero thickness and undergo prescribed flapping, as shown in Figs. 2 and 3, with a period of \(T\) = 0.0365 \(\mathrm {s}\). Figure 4 shows the contact point position along the leading edge over a flapping cycle. The position is measured from the body.

Fig. 1
figure 1

Wing dimensions

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Fig. 2
figure 2

Wing configurations at \(t/T\) \(=\) 0.0, 0.1, 0.2, 0.3, 0.4 and 0.5 (left to right and then top to bottom)

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Fig. 3
figure 3

Wing leading edges at the same instants as in Fig. 2

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Fig. 4
figure 4

Contact point position along the leading edge over a flapping cycle

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Fig. 5
figure 5

Computational domain and mesh setup. Outer boundaries (gray), boundaries of the inner, structured meshes (blue), and body (green). (Color figure online)

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The density and kinematic viscosity are \(1.225~\mathrm {kg}/\mathrm {m}^3\) and \(1.461\times 10^{-5}~\mathrm {m}^2/\mathrm {s}\). The free-stream velocity is \(4.5~\mathrm {m}/\mathrm {s}\). The Reynolds number based on average chord length and free-stream velocity is 5,423. Three cases are computed, with the angle of attack \(\alpha =0^{\circ }, 5^{\circ }\) and \(10^{\circ }\). The dimensions of the computational domain, in spans of a single wing, are 30 \(\times \) 20 \(\times \) 20, and the distance between the inflow boundary and the leading edge is \(10\) (see Fig. 5). The boundary conditions are no-slip on the wings and body, uniform horizontal velocity at the inflow boundary, zero-stress at the outflow boundary, and slip at the upper, lower and side boundaries.

The meshes have structured, inner zones around the wings, and an unstructured, outer zone. Both the structured and unstructured zones are made of tetrahedral elements. Table 1 shows the number of nodes and elements in the meshes used. Figure 6 shows top view of the wing and body surface meshes. During the flapping motion, only the mesh in the inner zones move, and this is done with a special, algebraic mesh moving technique.

Fig. 6
figure 6

Surface mesh at \(t/T=0.5\)

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Table 1 Number of nodes (nn) and elements (ne) in the meshes used
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The structured, inner zones for each wing consist of four parts. Those parts each have \(3\times 2\times 2\) structured zones. Each zone has \(20\times 20\times 20\) hexahedral clusters made of 6 tetrahedral elements. Figures 7 and 8 show, for \(\alpha =0^{\circ }\), the mesh at the same instants as in Fig. 2. The zones between the upper and lower wings collapse when the wings close, and the nodes in the neighboring zones also collapse accordingly. We note that a wing has split nodes except along its edges not attached to the body. However, when the wings are closed, the nodes on the upper surface of the upper wings and the lower surface of the lower wings become masters. When the wings are partially closed, at the contact point, the nodes on the lower surface of the upper wing are also masters while the nodes on the upper surface of the lower wings are slaves.

Fig. 7
figure 7

Mesh (cut mid-chord) at the same instants as in Fig. 2

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Fig. 8
figure 8

Mesh (cut mid-span) at the same instants as in Fig. 2

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2.2 Computational conditions

We use the ST-SUPS and ST-VMS (convective form) methods for the first two and last two nonlinear iterations of each time step, essentially making the ST-VMS method the operative one. The ST-SUPS method is the original DSD/SST method. It was named “DSD/SST-SUPS” in [10] (i.e. the version with the SUPG/PSPG stabilization), and gained the shorter name “ST-SUPS” in [12]. The ST-VMS method [11] is the the variational multiscale version of the DSD/SST method, which was first called “DSD/SST-VMST” (i.e. the version with the VMS turbulence model) in [10]. The VMS components are from the residual-based VMS method given in [76–79]. In these methods, the stabilization parameter \(\tau _{\tiny \mathrm{SUPS}}\) comes from the \(\tau _{\tiny \mathrm{SUPG}}\) definition in [8], specifically the definition given by Eqs. (107)–(109) in [8], which can also be found as the definition given by Eqs. (7)–(9) in [9], with \(\nu _{\tiny \mathrm{LSIC}}\) from Eq. (17) in [9].

The time-step size is \(4.51\times 10^{-4}~\mathrm {s}\). At each angle of attack, prior to the flapping motion, we compute 550 time steps with the geometry at \(t=0\) to develop the flow field. For the first 500 time steps, only half of the computational domain is used, with slip boundary condition on the symmetry plane. The computed data is then copied to the other half of the mesh for the final 50 time steps of flow field development. The inflow velocity \(4.5~\mathrm {m}/\mathrm {s}\) is reached by a sinusoidal ramping over the first 150 time steps, starting from \(0.0~\mathrm {m}/\mathrm {s}\). In computing the developed flow field, the number of GMRES iterations per nonlinear iteration is 150, 350, 450 and 800. In computing the flapping cycles, the number of GMRES iterations is 250, 500, 750 and 1,000. We compute three flapping cycles and display the results for the third cycle.

2.3 Results

We first present (in Figs. 9, 10, 11, 12, 13, 14), only for \(\alpha =10^{\circ }\), results over (or in relationship to) the MAV body and wing surfaces. Figure 9 shows the helicity isosurfaces. The flow field near the wings is almost symmetric, but the flow behind the MAV is not. Figures 10 and 11 show pressure on the body and the wing surfaces. The pressure is almost symmetric, and therefore we use the left and right sides of the wing pictures for the upper and lower wings. For the body, however, both sides show the upper surface. Figures 12 and 13 show the magnitude of shear stress on the body and wing surfaces.Again, we use the left and right sides of the wing pictures for the upper and lower wings, and both sides of the body for the upper surface. Figure 14 shows the pressure difference between the upper and lower surfaces. The left sides of the wing pictures are for the upper wing, and the right sides for the lower wing. For the closed parts of the wings, both sides show the difference between the lower surface of the lower wing and the upper surface of the upper wing. Lift and drag are shown in Figs. 15, 16 and 17. We also show in those figures the contributions to the lift and drag from the upper and lower wings, the closed wings, and the body.

Fig. 9
figure 9

Helicity isosurfaces (\(\pm 5\) and \(\pm 10\) \(\mathrm {m}^2/\mathrm {s}^2\)) for \(\alpha \) = \(10^{\circ }\) at \(t/T\) = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9 (left to right and then top to bottom). Blue is for negative values, and red for positive. (Color figure online)

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Fig. 10
figure 10

Pressure (\(\mathrm {Pa}\)) for \(\alpha =10^{\circ }\) on the body and wing surfaces at the same instants as in Fig. 9. The upper surface of the upper wing (left side) and the lower surface of the lower wing (right side). (Color figure online)

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Fig. 11
figure 11

Pressure (\(\mathrm {Pa}\)) for \(\alpha =10^{\circ }\) on the body and wing surfaces at the same instants as in Fig. 9. The lower surface of the upper wing (left side) and the upper surface of the lower wing (right side). The white regions are the closed parts of the wings. (Color figure online)

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Fig. 12
figure 12

Magnitude of the shear stress (\(\mathrm {Pa}\)) for \(\alpha =10^{\circ }\) on the body and the wing surfaces at the same instants as in Fig. 9. The upper surface of the upper wing (left side) and the lower surface of the lower wing (right side). (Color figure online)

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Fig. 13
figure 13

Magnitude of the shear stress (\(\mathrm {Pa}\)) for \(\alpha =10^{\circ }\) on the body and the wing surfaces at the same instants as in Fig. 9. The lower surface of the upper wing (left side) and the upper surface of the lower wing (right side). The white regions are the closed parts of the wings. (Color figure online)

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Fig. 14
figure 14

Pressure difference (\(\mathrm {Pa}\)) between the lower and upper surfaces for \(\alpha =10^{\circ }\) at the same instants as in Fig. 9. The upper wing and closed wings (left side) and the lower wing and closed wings (right side). (Color figure online)

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Fig. 15
figure 15

Lift (top) and drag (bottom) for \(\alpha =0^{\circ }\). (Color figure online)

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Fig. 16
figure 16

Lift (top) and drag (bottom) for \(\alpha =5^{\circ }\). (Color figure online)

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Fig. 17
figure 17

Lift (top) and drag (bottom) for \(\alpha =10^{\circ }\). (Color figure online)

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3 Concluding remarks

We have extended the ST-TC method to 3D computational analysis of flapping-wing aerodynamics of an MAV with wing clapping. The ST-TC method is a new version of the DSD/SST method, which is an interface-tracking (moving-mesh) method. The ST-TC method possess the desirable features of moving-mesh methods, including better resolution of the boundary layers, which is crucial in accurate computational analysis of flapping-wing aerodynamics. In its ST-TC version, the DSD/SST method is enhanced with a master–slave system that maintains the connectivity of the parent fluid mechanics mesh when there is contact between the moving interfaces. With that enhancement and because of its ST nature, the ST-TC method can deal with an actual contact between solid surfaces in flow problems with moving interfaces. It accomplishes that while still possessing the desirable features of moving-mesh methods, the key desirable feature being better resolution of the boundary layers. Even before its ST-TC version, the DSD/SST method, with effective mesh update, was already able to handle moving-interface problems when the solid surfaces are in near contact or create near TC. Flapping-wing aerodynamics of an actual locust, with the forewings and hindwings crossing each other very close and creating near TC, is an example of successfully computed problems. Flapping-wing aerodynamics of an MAV with the wings of an actual locust is another example. In computational analysis of flapping-wing aerodynamics with wing clapping, however, we need to bring the wings into an actual contact when they clap. We showed here that with the ST-TC method we can do that. We used a model dragonfly MAV in the computational analysis, with the wings brought into an actual contact when they clap. The work presented shows that the ST-TC method has the accuracy of moving-mesh methods, flexibility of being able to deal with the TC in the fluid mechanics mesh when the wings are brought into contact, and the computational practicality of accomplishing these in 3D analysis.