Keywords

1 Introduction

Over the past few decades, concerted efforts have been made by researchers to unravel the physical phenomena underlying the insects’ fascinating flight performances. Flapping-wing flight appears to be more advantageous in terms of its maneuverability and efficiency compared to conventional fixed-wing flight, especially for small size flyers. It is believed that successful application of insects’ aerodynamics could revolutionize the design of Micro Air Vehicle (MAV).

The earliest study of flapping wing aerodynamics attempted to estimate the forces generated by the wings using quasi-steady aerodynamics of conventional fixed-wing flyers [1,2,3,4,5]. Experimental biomechanics and fluid dynamics have played a key role in the study of flapping flight since the 1920s. The experimental studies by Willmott & Ellington [6, 7], Dickinson et al. [8], Fry et al. [9] and others have measured the wing kinematics and explored the unsteady aerodynamics related to flapping wing flight in great details. Computational Fluid Dynamics (CFD) arguably offers a complimentary and sometimes more expedient approach to access the unsteady aerodynamics at the scales of the insects. Benefiting from the continuously improving computing hardware and technology, the development of CFD has advanced rapidly in the past several decades. Full-fidelity CFD-FSI (computational fluid dynamics with fluid structure/body interaction) simulation is one promising approach to study the free flight of a flapping-wing insect, where the interactions between flyer and flow can be well resolved in a coupled manner. Wu et al. [10] successfully simulated the controlled free hovering flight of a model fruit fly with all six degrees of freedom, and investigated its ‘real-time’ aerodynamics and the role of active feedback control. Yao and Yeo [11, 12] subsequently extended the hovering model of Wu et al. to study the free longitudinal flights and simple manoeuvres (saccadic yawing and rapid sideslipping) of the fruit fly. Yao and Yeo [13, 14] adopted a similar numerical framework to study the hovering and the forward flight and sideslip manoeuvre of a hummingbird hawkmoth, wherein the significant reciprocating wing mass of the hawkmoth was also accounted for.

In this paper, we shall be concerned with simulating the forward flight of a model hummingbird hawkmoth, Macroglossum stellatarum, where the effect of wing mass and wing elevation motion is studied. The simulation of insects in free flight remains highly challenging due to their dynamical complexities and high computational cost. To accelerate the computation, two parallelization techniques are used in current simulation, i.e., open multi-processing (OpenMP) and graphics processing units (GPU) acceleration.

Section 2 gives the morphological details of the model hawkmoth and the basic methodology for the numerical simulation. Section 3 presents the results for the forward flight of the model hawkmoth. The key conclusions from present study are summarized in Sect. 4.

2 Methodology

2.1 Governing Equations and Numerical Discretization

The hummingbird hawkmoth (Macroglossum stellatarum) is adopted as the model flapping wing flyer in the present work. It is a good representative of the family Sphingidae, which usually exhibits superior flight performance among flying insects with relatively high Reynolds number (Re ≈ 3000). The flapping wing motion of hummingbird hawkmoth has a frequency (f) of approximately 70 Hz and a stroke amplitude (Φ) of about 115° according to available literature [15]. The model insect has a wing length (L) of 20.2 mm and total mass (M) of 0.21g, where the wings’ mass takes about 4.66%.

Wing motion is specified by three flapping angles: the stroke/sweep angle (ϕ), the elevation angle (θ) and the twist angle (ψ). The wing angles ϕ, θ, ψ are basically the Euler angles. They specify the action of the wings during flight and are thus prescribed functions or actively-controlled function of time t. Low-order Fourier series comprising simple sinusoidal functions are used to describe the wing kinematics of the present model flyer. The wing angle functions may thus be represented as follows:

$$ \left\{ \begin{aligned} & \phi (t) = - \frac{\Phi }{2}\cos (2\pi t) \\ & \theta (t) = \theta_{0} + \theta_{a1} \cos \left( {2\pi t} \right) + \theta_{b1} \sin \left( {2\pi t} \right) + \theta_{a2} \cos \left( {4\pi t} \right) + \theta_{b2} \sin \left( {4\pi t} \right) \\ & \psi (t) = \frac{\pi }{2} - \frac{\pi }{4}\sin (2\pi t) \\ \end{aligned} \right. $$
(1a-c)

where t is the non-dimensional time. The \({\theta }_{a}\) and \({\theta }_{b}\) are parameters that define the shape of wing tip trajectory, such as U-shaped, oval-shaped and ∞-shaped paths.

The fluid flow around the model insect is governed by the three-dimensional incompressible Navier-Stokes equations, in the non-dimensional Arbitrary Lagrangian-Eulerian (ALE) form given here:

$$ \left\{ \begin{aligned} & \nabla \cdot {\mathbf{u}} = 0 \\ & \partial_{t} {\mathbf{u}} = - ({\mathbf{u}} - {\mathbf{u}}^{g} ) \cdot \nabla {\mathbf{u}} + \frac{1}{{\text{Re}}}\nabla^{2} \,{\mathbf{u}} - \nabla p \\ \end{aligned} \right. $$
(2a-b)

where u and p represent the velocity and pressure field of the fluid domain respectively, and ug is the convection velocity of the computational node. A projection-based method is adopted to solve above equations, where the insect body and wings are modelled as non-slip surfaces Γ(t), on which \({\textbf{u}}({\textbf{x}})={\textbf{u}}^{{g}}({\textbf{x}})\). The pressure boundary condition is given by \({\textbf{n}} \cdot \nabla {\text{p}} = - {\textbf{n}} \cdot {\textbf{a}}\), where a is the acceleration of the surface node. Neumann-type boundary conditions are applied at the far field boundaries of the flow domain.

The free flight of the model insect is governed by Newtonian dynamics. Assuming the body and wings to be essentially rigid, and the wings to have negligible mass, the kinematic and dynamic equations for flight are given by:

$$ \left\{ \begin{aligned} & \frac{{\text{d}}}{{{\text{d}}t}}{\mathbf{X}}_{B} (t) = {\mathbf{V}}_{B} (t) \\ & \frac{{\text{d}}}{{{\text{d}}t}}{{\varvec{\Theta}}}(t) = \left[ {{\mathbf{K}}({{\varvec{\Theta}}})} \right] \cdot {{\varvec{\upomega}}}(t) \\ & \frac{{\text{d}}}{{{\text{d}}t}}\left( {M \cdot {\mathbf{V}}_{B} } \right) = - Mg{\mathbf{k}}_{g} + {\mathbf{F}}_{A} \\ & \frac{{\text{d}}}{{{\text{d}}t}}\left( {{\mathbf{\mathop{I}\limits^{\leftrightarrow} }}^{B} \cdot {{\varvec{\upomega}}}} \right) = {\mathbf{T}}_{A}^{B} \\ \end{aligned} \right. $$
(3a-d)

where XB(t) is the position of body centre and also the centre of mass (CoM) at time t; \(\boldsymbol{\Theta }\left( t \right)\) the orientation vector of the flyer and \(\left[ {{\textbf{K}}\left( \boldsymbol{\Theta } \right)} \right]\) the transformation matrix relating the rate of change of \(\boldsymbol{\Theta} (t)\) to the angular velocity ω(t) of the body. The \({\mathbf{\mathop{I}\limits^{\leftrightarrow} }}^{B}\) and \({\mathbf{\mathop{I}\limits^{\leftrightarrow} }}^{B} \cdot {{\varvec{\upomega}}}\) denote the inertia tensor and the angular momentum of the flyer about its CoM. The \({\textbf{F}}_{A} (t)\) and \({\textbf{T}}_{A}^{B} (t)\) denote the net resultant aerodynamic force and moment respectively acting on the flyer about CoM, which are obtained from the flow field solver. The closed form expressions for the dynamics with wing mass are rendered highly complex because of the complex kinematics of the two wings, plus the relative shifting of the insect’s CoM within the body frame. The reader is referred to [16] for its derivation and further details.

The configuration of the mesh used in current simulation is shown in Fig. 1. A uniform Cartesian background grid had been adopted for the domain – Fig. 1b gives a partial view to highlight the placement of the model fly. The whole mesh system consists of 4013 Cartesian background grids and around 60000 moving nodes near the model insect.

Fig. 1.
figure 1

Grid system for numerical simulation.

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The complex fluid-structure interaction (FSI) implies the need to iterate the solution process between flow solver \({\textbf{F[}}\Gamma {\text{(t)]}}\) and the dynamic solver \({\mathbf{S}}\left\{ {{\textbf{F}}\left[\Gamma \right]} \right\}\) of Eq. (3) at each time step to determine the updated/new configuration of the flyer Γ(t). The iteration is carried at each time step until the change between consecutive estimates of Γ is smaller than a given tolerance.

2.2 Projection Method

Projection method is applied to solve Eq. (2), which is further decomposed into Eq. (4a) and Eq. (4b). The pressure Poisson Eq. (4c) is then obtained by taking divergence of Eq. (4b) and invoking the continuity equation.

$$ \left\{ \begin{aligned} & \frac{{{\mathbf{u}}^{*} - {\mathbf{u}}^{n} }}{\Delta t} = \frac{1}{2}\left\{ \begin{gathered} \left[ { - ({\mathbf{u}} - {\mathbf{u}}^{g} ) \cdot \nabla {\mathbf{u}} + \frac{1}{{\text{Re}}}\nabla^{2} {\mathbf{u}}} \right]^{n + 1} \hfill \\ + \left[ { - ({\mathbf{u}} - {\mathbf{u}}^{g} ) \cdot \nabla {\mathbf{u}} + \frac{1}{{\text{Re}}}\nabla^{2} {\mathbf{u}} - \nabla p} \right]^{n} \hfill \\ \end{gathered} \right\} \\ & \frac{{{\mathbf{u}}^{n + 1} - {\mathbf{u}}^{*} }}{\Delta t} = - \frac{1}{2}\nabla p^{n + 1} \\ & \nabla^{2} p^{n + 1} = \frac{2}{\Delta t}\nabla \cdot {\mathbf{u}}^{*} \\ \end{aligned} \right. $$
(4a-c)

Important steps of the implementation are summarized below.

  • STEP 1. Compute intermediate velocity field \({\mathbf{u}}^{*}\)

    For interior nodes: solve Eq. (4a).

    For solid boundary nodes: \({\mathbf{u}}^{*}={\mathbf{u}}^{n+1}={\mathbf{u}}^{g,n+1}\). This is the non-slip velocity boundary condition where the fluid velocity at a boundary node is equal to the velocity of the boundary node itself.

  • STEP 2. Solve the pressure Poisson equation.

    For interior nodes: solve Eq. (4c).

    For solid boundary nodes: \(\mathbf{n}\cdot \nabla {p}^{n+1}=\mathbf{n}\cdot \left(-{\textbf{a}}_{b}+\frac{1}{Re}{\nabla }^{2}{\mathbf{u}}^{n+1}\right)\), where \({\textbf{a}}_{b}\) is the boundary node acceleration.

    For far field boundary nodes: \(\mathbf{n}\cdot \nabla {p}^{n+1}=0\), which is again a Neumann-type boundary condition.

  • STEP 3. Compute the final velocity field \({\mathbf{u}}^{n+1}\).

    For all computational nodes: solve Eq. (4b).

Above three steps are repeated until the solution converges. Any term containing the del operator (\(\nabla \)) or Laplace operator (\({\nabla }^{2}\)) requires spatial discretization, the scheme depends on the type of computation nodes. Standard procedures can be followed for the 7-point central difference scheme, SVD-GFD scheme will be elaborated in next section. STEP 1 and STEP 3 are straightforward, where the calculation can be done separately on each node. STEP 2 is more complicated because pressure needs to be updated simultaneously to the next time step for all nodes. A large linear system in the form of \(\mathbf{A}\mathbf{x}=\mathbf{b}\) is derived. A is a sparse matrix containing the coefficients of the spatial discretization. x is the pressure vector, the dimension of which is the total number of computation nodes. b stores the results of \(\frac{2}{\Delta t}\nabla \cdot {\mathbf{u}}^{*}\) and the boundary condition in STEP 2. In present study, the BICGStab method is adopted to solve the large linear system.

2.3 SVD-GFD Scheme

The generalized finite difference (GFD) method is based on the Taylor series expansion. Taylor series can represent an arbitrary function \(f\left(\mathbf{x}\right)\) as an infinite sum of terms calculated by the function’s derivatives at a single point. Equation (5) shows the Taylor series expansion of \(f\left(\mathbf{x}\right)\) at point \(\mathbf{x}={\mathbf{x}}_{0}\) up to the nth order. The value of the function at \(\mathbf{x}={\mathbf{x}}_{0}+\Delta \mathbf{x}\) can be approximated by the Taylor series at \(\mathbf{x}={\mathbf{x}}_{0}\). In solving the NS equations, the function \(f\left(\mathbf{x}\right)\) can be the pressure field \(p\left(\mathbf{x}\right)\) or any velocity component \(u\left(\mathbf{x}\right), v\left(\mathbf{x}\right), w\left(\mathbf{x}\right)\).

$$ f({\mathbf{x}}) = f({\mathbf{x}}_{0} ) + \sum\limits_{{1 \le i_{1} + i_{2} + i_{3} \le n - 1}} {\frac{{\Delta x^{{i_{1} }} \Delta y^{{i_{2} }} \Delta z^{{i_{3} }} }}{{i_{1} !i_{2} !i_{3} !}}\left[ {\frac{{\partial^{{i_{1} + i_{2} + i_{3} }} }}{{\partial x^{{i_{1} }} \partial y^{{i_{2} }} \partial z^{{i_{3} }} }}f} \right]}_{{{\mathbf{x}}_{0} }} + O(|\Delta {\mathbf{x}}|^{n} ) $$
(5)

Second order spatial accuracy can be maintained if the Taylor series is truncated after the 4th order derivatives (\(n=4\)). In this way, the first 19 derivatives are retained. If the values of function \(f\left(\mathbf{x}\right)\) are known on 19 surrounding nodes of \({\mathbf{x}}_{0}\), we can obtain 19 equations about the derivatives \(\frac{{\partial }^{{i}_{1}+{i}_{2}+{i}_{3}}}{\partial {x}^{{i}_{1}}\partial {y}^{{i}_{2}}\partial {z}^{{i}_{3}}}f\) at \({\mathbf{x}}_{0}\), which forms a closed linear system for solving the derivatives. However, practice shows that the 19 × 19 matrix tends to be ill-conditioned, which is mainly caused by the poor spatial arrangement of the surrounding nodes. A systematic nodal selection scheme is applied to find out the most suitable supporting nodes to form the equation system. Details of the scheme can be found in the work by Ang [17], Zhang [18] and Yao [19], which will not be elaborated here. The nodal selection scheme helps to improve the quality of the matrix but cannot eliminate the ill-conditioned problem. An over-determined algebraic system is used to solve this problem, which is formed by including more supporting nodes. Hence the final linear system is shown as follows, where N > 19 is the number of surrounding nodes.

$$ {\mathbf{S}}_{N \times 19} \partial {\mathbf{f}}_{19 \times 1} = \Delta {\mathbf{f}}_{N \times 1} $$
(6)
$$ \Delta {\mathbf{f}}_{N \times 1} = \left[ {\begin{array}{*{20}c} {f_{1} - f_{0} } & {f_{2} - f_{0} } & {...} & {f_{N} - f_{0} } \\ \end{array} } \right]^{T} $$
$$ {\mathbf{S}}_{N \times 19} = \left[ {\begin{array}{*{20}c} {\Delta x_{1} } & {\Delta y_{1} } & {\Delta z_{1} } & {0.5\Delta x_{1}^{2} } & {0.5\Delta y_{1}^{2} } & {0.5\Delta z_{1}^{2} } & \cdots & {\Delta x_{1} \Delta y_{1} \Delta z_{1} } \\ {\Delta x_{2} } & {\Delta y_{2} } & {\Delta z_{2} } & {0.5\Delta x_{2}^{2} } & {0.5\Delta y_{2}^{2} } & {0.5\Delta z_{2}^{2} } & {} & {\Delta x_{2} \Delta y_{2} \Delta z_{2} } \\ \vdots & {} & {} & {} & {} & {} & \ddots & \vdots \\ {\Delta x_{N} } & {\Delta y_{N} } & {\Delta z_{N} } & {0.5\Delta x_{N}^{2} } & {0.5\Delta y_{N}^{2} } & {0.5\Delta z_{N}^{2} } & \cdots & {\Delta x_{N} \Delta y_{N} \Delta z_{N} } \\ \end{array} } \right] $$
$$ \partial {\mathbf{f}}_{19 \times 1} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{\partial f_{{\left( {{\mathbf{x}}_{0} } \right)}} }}{\partial x}} & {\frac{{\partial f_{{\left( {{\mathbf{x}}_{0} } \right)}} }}{\partial y}} & {\frac{{\partial f_{{\left( {{\mathbf{x}}_{0} } \right)}} }}{\partial z}} & {\frac{{\partial^{2} f_{{\left( {{\mathbf{x}}_{0} } \right)}} }}{{\partial x^{2} }}} \\ \end{array} } & {\begin{array}{*{20}c} {\frac{{\partial^{2} f_{{\left( {{\mathbf{x}}_{0} } \right)}} }}{{\partial y^{2} }}} & {\frac{{\partial^{2} f_{{\left( {{\mathbf{x}}_{0} } \right)}} }}{{\partial z^{2} }}} & \cdots & {\frac{{\partial^{3} f_{{\left( {{\mathbf{x}}_{0} } \right)}} }}{\partial x\partial y\partial z}} \\ \end{array} } \\ \end{array} } \right]^{T} $$

Pseudoinverse matrix of \({\mathbf{S}}_{N\times 19}\) needs to be computed to solve for \(\partial {\mathbf{f}}_{19\times 1}\), which is accomplished by the approach of single value decomposition (SVD).

2.4 Computation Acceleration

Above shows a brief introduction for the computational framework for current simulation, more details can be referred to in [16]. So far, we can see the numerical simulation is very computationally intensive due to the large mesh scale and time-dependent iterative FSI solver and thus parallelization is essential in the computation. Two parallelization techniques are used in current simulation, i.e., open multi-processing (OpenMP) and graphics processing units (GPU) acceleration. OpenMP is a shared-memory parallel architecture which provides great flexibility to achieve parallelization, where minor change is required on the code based on serial computation. General-purpose computing on graphics processing units (GPGPU) is an emerging heterogeneous computing technique that performs massive parallelization on graphics processing unit (GPU). This technology is designed to achieve high float point operation rates and is suitable for acceleration of numerically intensive CFD computations. Nowadays, a general-purpose GPU is designed with thousands of processing units to achieve high float-point operation rates, the structure of which is very suitable for current CFD simulation on a large grid system. The calculation of the projection method (BICGStab algorithm) with GPU acceleration is much faster than that with CPU parallelization, and the GPU acceleration becomes more advantageous as the grid amount increases [20]. The main drawback of the GPU acceleration is programming complexity, especially when extensive calculation is required on a single mesh grid, such as the singular value decomposition (SVD) procedure in current study. SVD calculation of the CFD solver was thus performed on CPUs with OpenMP parallel computation in present simulations. Hence, we combine the above two parallelization techniques to utilize their respective advantages.

3 Results and Discussion

The forward flight can be divided into two main consecutive stages, body pitching down and continuous acceleration. The first stage can be accomplished within few wing cycles and does not affect the flight stability much, while we have to stabilize the pitching angle in the second stage. Hence, the first step to fly forward is to pitch down so the aerodynamic force in the horizontal direction is increased and body drag is reduced.

The pitching down process is governed by two factors, aerodynamic moment and wing inertia-induced moment. The normal wing motion is restricted within the stroke plane for simple and stable flight control. While in the nature, insects apply the elevation angle very often, which results in a wide variety of wing kinematics. The oval-shape wing motion is adopted here for rapid pitch control, which is demonstrated in Fig. 2. Such wing motion is governed by the wing elevation function shown by Eq. (1b). The wing sweeps lower than the wing root during the downstroke and higher than the wing root during the upstroke, so the wing tip trajectory is an oval shape. Function of this type of elevation motion is shown by Eq. (7), where θa is prescribed as 10° during the pitching down process.

$$ \theta = \theta_{a} \sin \left( {2\pi t} \right) $$
(7)

Both aerodynamic drag force and wing inertia force are in the opposite direction of wing motion. Wing drag/inertia force in downstroke has shorter moment arm than that in upstroke, which contributes a net pitching down aerodynamic/inertia-induced moment in a wingbeat. The magnitude of both moments increases with larger elevation amplitude. The aerodynamic moment superposes with the wing inertia-induced moment, and together they make the insect pitch down rapidly.

Fig. 2.
figure 2

Oval-shape wing motion

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Since wing inertia-induced force is the internal force between the wings and body, the body pitch up (down) caused by wing mass does not impose pitching velocity or acceleration, governed by the law of conservation of angular momentum. However, that is unavoidable in pitch motion due to the aerodynamic moment. Hence, orientation adjustment by wing inertia is theoretically more straightforward and easier to control.

After the fast pitching down stage, we need to maintain the designated pitching angle to ensure the insect fly forward stably. When the insect starts to gain speed, the pitching up moment induced by the flow field increases for any wing plane angle values, which can be seen from the prescribed motion study shown in Fig. 3. In that case, we maintain the oval wing motion to eliminate the induced moment. This was also discussed by Yao and Yeo [11].

Fig. 3.
figure 3

Pitching moment variation with forward speed at different wing plane angles

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As indicated by Fig. 3, the induced pitching up moment increases almost linearly with the speed at different wing plane angles. Therefore, we can simply relate the wing elevation amplitude and forward speed linearly as in Eq. (8).

$$ \theta_{a} = k^{\theta } v $$
(8)

A constant linear coefficient may not be able to sustain a stable flight even though the pitching moment generated by oval wing motion may change linearly with elevation amplitude. The reason is because we cannot maintain the pitching angle at the designated value throughout the flight, and the actual induced moment may not follow exactly the trend shown in Fig. 3. Hence, we need modification to this linear relationship. Alternating linear coefficient can be used to counter the uncertain induced moment. If the insect tends to pitch up due to positive moment, we apply a larger coefficient to achieve a stronger pitching down effect. Similarly, we apply a smaller coefficient if it tends to pitch down. The alternating coefficient has advantage over a constant coefficient that it can stop the excessive moment from building up with a stronger recovery in time.

As analyzed previously, wing mass plays a significant role in pitch control during forward flight. To verify the analysis and find out the effect of wing mass, we simulated the forward flight in two situations, i.e., with or without wing mass. Since the effect of wing mass is mainly on the longitudinal motion, simulations reported in this paper do not take into consideration of the lateral dynamics to avoid the influence on each other. Figure 4 shows the comparison of forward velocity between flight with actual wing mass ratio and that without wing mass, where the wingbeat frequency is fixed at 75 Hz. Clearly, we can find that the case without wing mass shows much smaller cyclic velocity oscillation. Same conclusion can be drawn for the pitching motion, which is presented in Fig. 6. Hence, greater body oscillation is observed due to the wing inertia force. The flight with zero wing mass ratio accelerates faster and reaches a slightly higher speed in the end. The reason for that is because smaller body oscillation allows longer wing sweeping distance, which generates more aerodynamic forces. This can be verified by the wing tip trajectory at maximum speeds for both cases (see Fig. 5). The vertical wing sweeping distance is less dependent on the forward speed compared to the longitudinal sweeping distance, and we find that the positive longitudinal force is mainly generated in the upstroke, during which the wings move almost vertically viewed from global frame. Figure 5a shows that the flight without wing mass has a slightly longer range of wing tip trajectory in the vertical direction (\({h}_{1}>{h}_{2}\)) due to smaller body oscillation. Hence, more longitudinal thrust is generated for the case without wing mass, which contributes to faster acceleration and higher final speed.

Pitch control is more rapid and responsive for the flight with wing mass, as can be observed from Fig. 6a. The period of medium-term pitching oscillation for the case with wing mass is shorter than that without. This is because the wing inertia-induced moment gives rise to additional body rotation, which was explained previously. Another major difference is in the pitching down stage at the beginning of acceleration (first 3 wing cycles). The flyer pitches down faster in the case with wing mass because this process is accomplished by both the aerodynamic moment and the wing inertia-induced moment, as shown in Fig. 6b. In addition, the residual pitching fluctuation after that is smaller since the internal wing inertia force does not change the pitching velocity. However, for the case without wing mass, the whole pitching down motion is contributed by aerodynamic moment alone, which can also change the pitching velocity.

Fig. 4.
figure 4

Comparison of speed for forward acceleration with/without wing mass

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Fig. 5.
figure 5

Wing tip trajectories for forward acceleration with/without wing mass. Figures are based on actual flight data. Wing root is marked as (+) in figure. Wing chords show the approximate wing orientation with dotted leading edge. Unit wing length is represented by L.

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Fig. 6.
figure 6

Comparison of pitch for forward acceleration with/without wing mass

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4 Conclusion

The present paper investigated the effect of wing mass and wing elevation motion during insect free forward flight via numerical simulation. The numerical model integrated a Navier-Stokes flow solver with the Newtonian free-body dynamics of the model insect. Parallel computation is essential for current study, since the free flight simulation is computationally intensive due to the large mesh scale and the iterative solution for the FSI problem. Two parallelization techniques are used in current simulation, i.e., open multi-processing (OpenMP) and graphics processing units (GPU) acceleration. Two techniques are applied at the suitable computational algorithms respectively to take advantages of each. It is found that Oval-shaped wing elevating motion can help to generate large pitching down moment, both in terms of the aerodynamic moment and the wing inertia-induced moment. With such wing elevation motion, the flyer can quickly adjust its orientation for forward acceleration from the normal hovering status. On the other hand, wing mass tends to magnify such effect while prohibits the growth of pitching down velocity, which is favourable for the overall flight performance. Larger body oscillation is observed during forward flight for the case with actual wing mass compared to the case without wing mass. In addition, the forward flight with zero wing mass ratio accelerates faster and reaches a slightly higher speed in the end compared to the case with insect actual wing mass. The present study provides detailed information and access to the coupled dynamics of fluid and flyer in free flight condition. The present free flight model offers a prospective approach that could complement existing experiments with live insect subjects in a wider study of ‘real-time’ dynamics of insect flight and manoeuvres.