Scaling laws for the sound generation of bio-inspired flapping wings during hovering flight

Bio-inspired flapping wings, as an important way of aquatic and aerial animals moving through air and water, have been shown to have superior aerodynamic and hydrodynamic performance in comparison to the stationary wings in the low Reynolds number ($O(10^{3})$ or lower) regime (see e.g., Lighthill, 1975; Sane, 2003; Wang, 2005; Wu, 2011; Shyy et al., 2013; Eldredge & Jones, 2019; Liu et al., 2024). The superior performance is achieved through several unique mechanisms such as delayed stall of leading edge vortex (LEV) (Ellington et al., 1996), wake capture (Dickinson et al., 1999; Bomphrey et al., 2017), rapid pitching rotation (Bennett, 1970; Dickinson et al., 1999), unsteady wing-tip vortex (Shyy et al., 2010), synchronised passive pitching deformation (Shyy et al., 2010; Nakata & Liu, 2012a, b; Dai et al., 2012; Tian et al., 2013; Huang et al., 2023), and clap and fling (Weis-Fogh, 1973; Lighthill, 1973). For aerial animals and bio-inspired aerial vehicles, the noise generated by flapping wings is also an important factor measuring the aerodynamic performance, as discussed in a few recent studies (Geng et al., 2017; Wang & Tian, 2019, 2020; Jaworski & Peake, 2020; Clark, 2021). The study of the sound generation of bio-inspired flapping wings would shed light on the sound control of bio-inspired aerial vehicles.

One of the most fascinating features of bio-inspired flapping wings is the high lift generation at low Reynolds numbers (e.g. $O(10^{2})$ to $O(10^{3})$) by using unsteady flow mechanisms (Shyy et al., 2010, 2013; Eldredge & Jones, 2019; Liu et al., 2024). The unsteady flow mechanisms are realised through the complicated kinematics which could be one or the combination of stroke, pitching, deviating, twisting and revolving components (Bomphrey et al., 2017). Different components contribute to the aerodynamic force generation in different ways (Shyy et al., 2010; Eldredge & Jones, 2019; Liu et al., 2024). For example, the stroke motion (also known as the translational motion for 2D cases) creates an intense LEV undergoing a delayed stall to generate high lift which is not possible according to conventional laws of aerodynamics (Ellington et al., 1996; Kweon & Choi, 2010). The stabilised LEV observed for low Reynolds number insects/birds flight has been further elaborated by using experimental measurements (Poelma et al., 2006) and direct numerical simulations (Liu et al., 1998; Liu & Aono, 2009), showing that the attached LEV generates low pressure regions enhancing the lift generation during translation. The pitching (rotational) motion of the wing generates the lift during stroke reversal through the Kramer effect and the wake capture (Dickinson et al., 1999; Sane & Dickinson, 2002). The wing revolving/rotating enhances the lift generation by stabilising the LEV through the spanwise pressure gradient, centripetal force, and Coriolis force (Ozen & Rockwell, 2012; Jardin & David, 2015). In addition, the dynamic bending of flexible wings delays the breakdown of LEV near the wing-tip augmenting the production of stroke-averaged vertical force (Nakata & Liu, 2012a).

Even though the unsteady kinematics could enhance the force production significantly, the force variations are also larger compared with that of the steady wings. The force variations may have significant consequences on the acoustics generated by the wings especially the Gutin sound (Clark, 2021), which have not been well explored (Nedunchezian et al., 2019; Wang & Tian, 2020). The flapping-wing induced sound has drawn the attention from zoologists in very early stages. For example, Ewing & Bennet-Clark (1968) found that the complex sound produced by drosophila has a similar frequency to the wing beat frequency during flight. One of the earliest rigorous study on the aeroacoustics of flapping wing can be originated to the experimental study conducted by Sueur et al. (2005) who measured the acoustic outputs of flies in tethered flight, finding that the first frequency component is associated with the wing beat frequency dominating the front of the flyer and the second harmonic is dominate at two sides. The two-dimensional numerical study conducted by Bae & Moon (2008) identified two different mechanisms of sound generation by flapping wings, where the frequency of the primary dipole tone is the same as the frequency of the transverse wing, while the vortex edge scattering during a tangential motion is responsible for other higher frequency dipole tones. Bae & Moon (2008) also pointed out that the wing-vortex effects are more prominent in forward flight resulting in a broader high frequency noise band. To further elaborate the sound generation by flapping wing in real world, three dimensional numerical studies have been conducted by Inada et al. (2009); Geng et al. (2017); Wang & Tian (2020). Specifically, the pressure variation on the wing surface is the major source of the acoustics generated by the flapping wing (Inada et al., 2009), which is consistent with the classical acoustics analogy theory. Geng et al. (2017) adopted a hydrodynamic/acoustic splitting approach to numerically study the sound generated by a forward-flying cicada. By using a prescribed wing kinematics considering the wing deformation, they found that the flexibility is beneficial in the noise reduction in all directions. Inspired by the numerical simulations considering fully flexible flapping wings which demonstrated the enhanced aerodynamic performance by appropriate flexibility, Wang & Tian (2019) considered the fluid–structure-acoustics interaction of flapping wings and found that an aerodynamics-dominated flexible wing experiences sound reduction and aerodynamic performance enhancement, while the flexibility of inertia-dominated wings strengths the sound outputs. Nedunchezian et al. (2019) numerically studied the sound generation by flapping wings with various kinematics with Ffowcs Williams-Hawkings (FW-H) equations and found that the high lift and the low sound are found with the large-amplitude flap and pitch, and the delayed pitching motion. Nedunchezian et al. (2019) also pointed out that the delayed pitching motion of the wing may aim to decrease the sound generation instead of enhancing the lift force. The three-dimensional fluid-structure-acoustics interaction considering three-dimensional flapping wings in hovering flight by Wang & Tian (2020) confirmed that the aerodynamics-dominated wing could experience noise reduction and lift enhancement simultaneously.

Even though scaling laws for the aerodynamics of flapping-wing have been developed (Dewey et al., 2013; Kang & Shyy, 2013; Lee et al., 2015; Quinn et al., 2014; Floryan et al., 2017; Sum Wu et al., 2019; Senturk & Smits, 2019; Ayancik et al., 2019), they have not been considered for the acoustics except for two recent works (Wang & Tian, 2020; Seo et al., 2019). Specifically, a simple scaling law for the noise output from flapping wings versus their aerodynamic power requirements was developed in Wang & Tian (2020). Seo et al. (2019) reported that the sound power efficiency is proportional to $f^{2}$. The present work is to develop scaling laws for the aeroacoustics of flapping wings during hovering flight based on the quasi-steady flow theory and Ffowcs-Williams-Hawkings analogy. Specially, we first consider purely translating wings in two-dimensional domain. We then consider two-dimensional flapping wings with a combined translating and pitching motion. Finally, we consider three-dimensional flapping wings during hovering flight. The rest of this work is organised as follows: the physical formation of flapping wing considered in this study is introduced in § 2; the numerical method used for the direct numerical simulation of the aerodynamics and aeroacoustics of flapping wing is introduced in § 3; the results and discussions are then presented in § 4; and a final conclusion is given in § 5.

To develop the scaling laws for hovering flapping wings, the sound generation by flapping wings in both two-dimensional (2D) and three-dimensional (3D) spaces is considered, as shown in Fig. 1. For the simplicity, the wing translating at a constant angle of attack namely $\frac{\pi}{2}-\alpha_{m}$ in the horizontal direction is first considered, as shown in Fig. 1 (a). The kinematics of the translation is prescribed by the following equation

$$x=A_{m}\sin(2\pi ft),$$

(1)

where $A_{m}$ is the amplitude of the translation, $f$ is the frequency, $x$ is the position of the leading edge and $t$ is the time. For the flapping wing with a combined translating and pitching motion, as shown in Fig. 1 (b), the pitching montion is prescribed as

$$\alpha=\alpha_{m}\cos(2\pi ft),$$

(2)

where ${\pi}/{2}-\alpha_{m}$ is the maximum angle of attack. The dimensionless parameters governing this problems include translating amplitude, $a_{m}$, Reynolds number and Mach number,

$$A_{m}/\bar{c},\quad a_{m},\quad Re=\frac{\rho U\bar{c}}{\mu},\quad M=\frac{U}{c_{s}},$$

(3)

where $U=2\pi fA_{m}$, $\bar{c}$ is the mean chord length, $c_{s}$ is the speed of sound, and $\rho$ and $\mu$ are respectively the density and viscosity of the fluid. In this work, low Reynolds numbers and Mach numbers are considered to limit the scope for insect flight, as shown in Tab. 1. Other parameters of kinematics are also selected based on insects. For the 3D flapping wing undergoing stroke and pitching motion, as shown in Fig. 1 (c), the kinematics is governed by

$$\phi=\phi_{m}\sin(2\pi ft),\quad\alpha=\alpha_{m}\cos(2\pi ft),$$

(4)

where $\phi_{m}$ and $\alpha_{m}$ are respectively the amplitude of stroke and pitching. The flow of the 3D hovering flapping wing is governed by $\phi_{m}$, $\alpha_{m}$, Reynolds number and Mach number as defined in Eq. 3 with $U=2\pi f\phi_{m}\bar{c}AR$, and the aspect ratio $AR$ of the wing. A range of parameters are examined to construct the scaling law for the sound generation, as shown in Tab. 1. Therefore, the fluid flow is considered as weakly-compressible. The lift and drag coefficients are defined as

$$C_{L}=\frac{F_{L}}{0.5\rho U^{2}A},\quad C_{D}=\frac{F_{D}}{0.5\rho U^{2}A},$$

(5)

where $F_{L}$ and $F_{D}$ are respectively the sum of force exerted by the fluid on the wing in the vertical and horizontal directions which can be directly obtained by using the sum of the Lagrangian force introduced in the immersed boundary method (IBM), and $A$ is the area of the wing which is $\bar{c}$ for 2D cases. The sound pressure is defined as $\Delta p=p-\bar{p}$ with $\bar{p}$ being the time averaged pressure, and the time averaging is conducted using ten cycles after the periodic state (cycle-to-cycle variation in the forces is negligible) is achieved. All the sound pressure presented here are scaled by the flow density and sound speed in quiescent state.

The unsteady aerodynamics of bio-inspired flapping wings is directly solved by using an immersed boundary–lattice Boltzmann method (IB-LBM), where the weakly compressible flow is solved by the LBM and the no-slip boundary condition is implemented by an IBM. Without loss of generality, here we use 2D IB-LBM as an example to introduce the method. In the multi-relaxation-time (MRT)-based LBM, the particle distribution function $g_{i}$ is given as (Coveney et al., 2002; Guo & Zheng, 2008; Xu et al., 2018)

$$g_{i}(\boldsymbol{x}+\boldsymbol{e}_{i}\Delta t,t+\Delta t)=g_{i}(\boldsymbol{x},t)-\Omega_{i}(\boldsymbol{x},t)+\Delta tG_{i},\quad i=1,2,...9,$$

(6)

where $\Delta t$ denotes the time step, $\boldsymbol{e}_{i}$ is the discrete velocity, subscript $i$ indicates the $i$-th direction, $\boldsymbol{x}$ is the space coordinator, $\Omega_{i}$ the collision operator, and $G_{i}$ is the effect of the body force. For the MRT LBM, $\Omega_{i}$ and $G_{i}$ are given as,

	$\displaystyle\Omega_{i}$	$\displaystyle=$	$\displaystyle-(M^{-1}SM)_{ij}[g_{j}(\boldsymbol{x},t)-g_{j}^{eq}(\boldsymbol{x},t)],$		(7)
	$\displaystyle G_{i}$	$\displaystyle=$	$\displaystyle[M^{-1}(I-S/2)M]_{ij}F_{j},$		(8)

where $M$ is a transform matrix with a dimension of $9\times 9$ for the 2D nine discrete velocity (D2Q9) model and $19\times 19$ for 3D nineteen discrete velocity (D3Q19), $S$ is the collision matrix, $I$ is the identity matrix, $g_{i}^{eq}$ is the equilibrium distribution function, and $F_{j}$ is the force effect term. Details of $M$ and $S$ can be found in Coveney et al. (2002). Here $g_{i}^{eq}$ and $F_{j}$ can be calculated by

	$\displaystyle g_{i}^{eq}=\omega_{i}\rho[1+\frac{\boldsymbol{e}_{i}\cdot\boldsymbol{u}}{c_{s}^{2}}+\frac{\boldsymbol{u}\boldsymbol{u}:(\boldsymbol{e}_{i}\boldsymbol{e}_{i}-c_{s}^{2}\boldsymbol{I})}{c_{s}^{4}}],$		(9)
	$\displaystyle F_{j}=\omega_{j}[\frac{\boldsymbol{e}_{j}-\boldsymbol{u}}{c_{s}^{2}}+\frac{\boldsymbol{e}_{j}\cdot\boldsymbol{u}}{c_{s}^{4}}\boldsymbol{e}_{j}]\cdot\boldsymbol{f},$		(10)

where $\omega_{i}$ are the weights, $c_{s}=\Delta x/(\sqrt{3}\Delta t)$ is the the sound speed, and $\boldsymbol{f}$ is the body force. The kinematic viscosity $\nu$ of the fluid is determined by the relaxation time $\tau$, according to $\nu=(\tau-0.5)c_{s}^{2}\Delta t$ (Qian et al., 1992; Chen & Doolen, 1998; Wang et al., 2022).

The macro variables including the density, pressure and momentum can be directly calculated from the distribution functions as follows,

$$\rho=\sum_{i=1}^{9}g_{i},\quad p=\rho c_{s}^{2},\quad\rho\boldsymbol{u}=\sum_{i=1}^{9}g_{i}\boldsymbol{e}_{i}+\frac{1}{2}\boldsymbol{f}\Delta t.$$

(11)

Here, the feedback immersed boundary method proposed (see e.g., Goldstein et al., 1993; Huang & Tian, 2019) is employed to handle the no-slip boundary conditions at the fluid–structure interface,

	$\displaystyle\boldsymbol{f}(\boldsymbol{x},t)=-\int_{\Gamma}\boldsymbol{F}_{L}(s,t)\delta_{h}(\boldsymbol{X}(s,t)-\boldsymbol{x})ds,$		(12)
	$\displaystyle\boldsymbol{F}_{L}=\alpha_{ib}\int_{0}^{t}(\boldsymbol{U}_{ib}-\boldsymbol{U})dt+\beta_{ib}(\boldsymbol{U}_{ib}-\boldsymbol{U}),$		(13)
	$\displaystyle\boldsymbol{U}_{ib}(s,t)=\int_{V}\boldsymbol{u}(x,t)\delta_{h}(\boldsymbol{X}(s,t)-\boldsymbol{x})d\boldsymbol{x},$		(14)

where $\boldsymbol{F}_{L}$ is the Lagrangian force, $\boldsymbol{U}_{ib}$ is the boundary velocity obtained by the interpolation from the fluid domain, $\alpha$ and $\beta$ are positive feedback constants, $\boldsymbol{u}$ is velocity of fluid nodes, $\boldsymbol{X}$ and $\boldsymbol{x}$ are respectively the coordinates of structural nodes and fluid nodes, $s$ is arc coordinate, $V$ denotes the fluid domain, $\Gamma$ denotes the structure domain, and $\delta_{h}$ is the smoothed Delta function. In this work, the delta function proposed by Peskin (Peskin, 2002) is used. In addition, $\alpha_{ib}=0$ and $\beta_{ib}=2.0$ are used. More details can be found in Refs. (Wang & Tian, 2018; Tian et al., 2011; Xu et al., 2018; Huang et al., 2022).

The computational domain shown in Fig. 2(a) extends respectively from $-150\bar{c}$ to $150\bar{c}$ in both directions for 2D and from $-50\bar{c}$ to $50\bar{c}$ in all three directions for 3D. A sponge layer is applied respectively at a distance of $100\bar{c}$ (2D) and $45\bar{c}$ (3D) for the outer domain to damp the solution to the far-field quiescent state, and a characteristic non-reflective boundary condition is also applied at the external boundaries to guarantee the reflections from the external boundaries are negligible. Seven (2D) and five (3D) levels of mesh are used in present study, and the mesh is refined with a factor of two with the finest mesh spacing of $0.005\bar{c}$ (2D) and $0.02\bar{c}$, respectively. Three mesh blocks close to the wing is shown in Fig. 2(b) to illustrate the mesh strategy. The finest mesh covers the sweeping range of the wing for the near-field flow, and the acoustics are far away from the wing is resolved with mesh resolution of $0.16\bar{c}$ namely the acoustics region in Fig. 2(a). The Lagrangian mesh spacing is about $0.5\Delta x$, as demonstrated in Fig. 2(c). In the current simulation, a reference density $\rho=1.0$ and a sound velocity $c_{s}=1/\sqrt{3}$ are used. The sound pressure is nondimensionalised by $\rho c_{s}^{2}$. The validation of the numerical solver is given in Appendix A, and the mesh convergence study is provided in Appendix B. It should be noted that the force and sound power are transformed into the physical space assuming a wing mean chord length of $2mm$ in earth atmosphere environment, to make it intuitive for readers.

This work presents the scaling laws for the sound generation of bio-inspired flapping wings during hovering flight, which are derived by using the classical quasi-steady flow theory and the FW-H acoustic equation. Three groups of wings are consider: 2D purely translating wings, 2D translating and pitching wings, and 3D stroking and pitching wings. Direct numerical simulations considering a range of parameters including the Reynolds number, Mach number and wing kinematics confirm that the constructed scaling laws capture the major physics and agree well with the direct numerical simulation results.

It is noted that the simple scaling laws directly relate the kinematics of bio-inspired flapping wings to their noise emission. The scaling laws are able to explain the properties of sound generated by most insects with different wing sizes and kinematics. The success of this work is due to the fact that the major sound by the bio-inspired flapping wings considered here is the Gutin sound. The scaling analysis of other sound sources should be considered for complicated wings and high-speed flight, which will be our future effort.

This work was supported by the Australian Research Council Discovery Project (grant No. DP200101500) and conducted with the assistance of computational resources from the National Computational Infrastructure (NCI), which is supported by the Australian Government.

The authors report no conflict of interest.