3.1. Time derivative of surface pressure force

The dominant contributor to the sound pressure in the far-field is the time derivative of the surface pressure $\dot {p}$ and thus the time derivative of surface pressure force $\dot {f_{p_i}}$ (see (2.8)). To discuss it, $\dot {f_{p_i}}$ over a flapping cycle $t/T=0$ to 1 ($T$ is the flapping period) is shown in figure 3(a). At $t/T=0.285$, $\dot {f_{p_i}}$ in the three directions is close to zero, which will be further investigated. On the other hand, the maximum fluctuation of $\dot {f_{p_i}}$ occurs near $t/T=0.42$ when $\dot {f_{p_x}}$ achieves its minimum value while the fluctuations of other two components are also high.

Figure 3.

Force properties and vortical structures: (a) time derivative of the pressure force in the three directions; (b) instantaneous dimensionless surface pressure at $t/T=0.285$; (c) instantaneous dimensionless time derivative of the surface pressure at $t/T=0.285$; (d) vortices visualised by $Q$-criterion ($Q=30$) and coloured by pressure at $t/T=0.285$; (e) instantaneous dimensionless surface pressure at $t/T=0.42$; ( f) instantaneous dimensionless time derivative of the surface pressure at $t/T=0.42$; (g) vortices visualised by $Q$-criterion ($Q=30$) and coloured by pressure at $t/T=0.42$.

The instantaneous surface pressure and its time derivative together with the vortices visualised by the $Q$-criterion (Jeong & Hussain Reference Jeong and Hussain1995) over the suction side at $t/T=0.285$ and $t/T=0.42$ are presented in figures 3(b–d) and 3(e–g), respectively. The $Q$-criterion is used here because it can be taken as the acoustics and the surface pressure variation source, as will be discussed in § 3.3. The negative pressure over the outboard region at $t/T=0.285$ is much larger than that at $t/T=0.42$ (see figure 3b,e) but no significant difference in the vortical structures between these two time steps can be identified (see figure 3d,g). In addition, a $\boldsymbol {\varPi }$-like distribution pattern with a high negative pressure ($p<-0.7$, see figure 3e) can be identified in the inboard region at $t/T=0.42$, which is associated with the low pressure within the LEV, RV and SV (SV1). In particular, the negative pressure near the wing root at $t/T=0.42$ is larger than that at $t/T=0.285$ due to the larger RV. At $t/T=0.285$, $\dot {p}$ over the outboard region is positive which compensates for the negative value over the inboard region, yielding a small amplitude $\dot {f_{p_i}}$ (see figure 3a,c). At $t/T=0.42$, $\dot {p}$ over the suction side is generally positive, indicating that the surface pressure increases, and a region R1 with a noticeably high value is found at around a chord from the wing root (see figure 3f). Although the value of $\dot {f_{p_i}}$ can be explained by the contours of $\dot {p}$, the relationship between the vortical structure evolution and $\dot {p}$ cannot be determined from figure 3. Furthermore, it should be noted that $\dot {p}$ generally increases from the wing root to the wingtip at $t/T=0.285$. However, R1 does not occur at the wingtip at $t/T=0.42$, which may contain some important physical insights and shall be investigated. The LEV, as a prominent vortical component, may significantly affect $\dot {p}$ and result in the high $\dot {p}$ at R1 (Eldredge & Jones Reference Eldredge and Jones2019). Therefore, the LEV circulation will be quantified and examined in the next section.

3.2. The LEV evolution

To investigate the correlation between the LEV and the high $\dot {p}$ of R1, a body-fixed reference frame $s$–$n$–$\tau$, with its origin at the corner closest to the pivot point of the wing (see figure 4a) is first established, where $s$ represents the direction from the trailing edge to the leading edge, $n$ denotes the normal to the wing surface and $\tau$ points from the wing root to the wingtip. The vorticity $\boldsymbol {\omega }=\boldsymbol {\nabla } \times \boldsymbol {u}$ is calculated in the $s$, $n$ and $\tau$ directions, respectively, and the circulation of the LEV, $\varGamma _{\tau }=\iint \omega _{\tau } \,{\rm d}A$, is defined here to quantify the vorticity strength of the LEV and its variation near R1 (see figure 3f). In figure 4(b), the integration area $A$ is defined as a rectangular region to exclude the contribution of the vorticity of SV1 (see figure 3d,g). Additionally, the LEV is bounded by $Q>0$ and $\omega _{\tau }<0$ (similar definition can be found in Chen et al. (Reference Chen, Wang, Zhou, Wu and Cheng2022)). Here, two slices S1 ($\tau =0.75c$) and S2 ($\tau =0.95c$) (see figure 4c,d), are chosen for calculating the LEV circulation. Here S1 is placed out of R1 while S2 crosses R1. It should be noted that the LEV will lose its coherence at the further outboard area where the tilting of the TV may affect the estimation of the LEV circulation. Therefore, the LEV circulation at the outboard region is not considered here. Apart from $\varGamma _{\tau }$, the vorticity fluxes $\hat {\omega }_s$ and $\hat {\omega }_n$ at S1 and S2 are obtained by integrating the other two vorticity components within the LEV region to show the strength of vorticity in the $s$ and $n$ directions.

Figure 4.

The LEV structures: (a) schematic of the body-fixed reference frame $s$–$n$–$\tau$ (marked by the red colour); (b) schematic of circulation integration area and the LEV is highlighted by the contours of $\omega _\tau$; (c) schematics of S1–3, with the contours of $\dot {p}$ at $t/T=0.285$; (d) schematics of S1–3, with the contours of $\dot {p}$ at $t/T=0.42$; (e) the LEV circulation $\varGamma _\tau$, and vorticity fluxes $\dot {\hat {\omega }}_s$ and $\dot {\hat {\omega }}_n$ of S1 and S2.

Overall, it can be observed from figure 4(e) that $\varGamma _\tau$ is larger with respect to the vorticity fluxes in the other two directions ($\hat {\omega }_s$ and $\hat {\omega }_n$) for both slices, and the values at S2 are generally larger than those at S1, suggesting that the LEV is more 3-D and unsteady near R1 over the inboard region. At S1, $\hat {\omega }_s$ and $\hat {\omega }_n$ have a similar variation tendency and share a similar maximum value of around 0.35 at $t/T=0.38$. At S2, the variation of $\hat {\omega }_s$ and $\hat {\omega }_n$ is different after achieving its maximum value at $t/T=0.33$. At $t/T=0.285$, the variation trend of $\varGamma _\tau$ at S1 and S2 is similar, which may explain why $\dot {p}$ has a similar magnitude near S1 and S2 (see figure 4c). Meanwhile, $\varGamma _\tau$ of S1 and S2 at $t/T=0.42$ increases, indicating a reduction of the LEV circulation, and the increase rate of $\varGamma _\tau$ at S2 is larger with respect to that of S1, corresponding to the higher $\dot {p}$ at R1 (see figure 4d). Moreover, the difference in the variation rate at $t/T=0.42$ between S1 and S2 is larger than that at $t/T=0.285$.

Additionally, the cross-correlations between $\dot {\varGamma }_\tau$, $\dot {\hat {\omega }}_s$, and $\dot {\hat {\omega }}_n$ and $\dot {p}$ are incorporated to investigate and quantify the relationship between the time derivatives of the pressure and vortex evolution,

(3.1a,b)\begin{align} R_{\dot{\varGamma}_\tau \dot{p}} = \frac{1}{N}\sum_{n=1}^{N} \frac{(\dot{\varGamma}_\tau - \overline{\dot{\varGamma}_\tau})(\dot{p} - \bar{\dot{p}})}{\sigma_{\dot{\varGamma}_\tau} \sigma_{\dot{p}}}, \quad R_{\dot{\hat{\omega}}_i \dot{p}} = \frac{1}{N}\sum_{n=1}^{N} \frac{(\dot{\hat{\omega}}_i - \overline{\dot{\hat{\omega}}_i})(\dot{p} - \bar{\dot{p}})}{\sigma_{\dot{\hat{\omega}}_i} \sigma_{\dot{p}}}, \quad i=n,s, \end{align}

where $(^{{\cdot }})$ denotes the time derivative, $N$ is the total sample number, $\dot {\varGamma }_\tau$ and $\dot {\hat {\omega }}_i$ are the time derivatives of the LEV circulation and vorticity fluxes of the LEV in the $n$ and $s$ directions at S2, $\overline {({\cdot })}$ denotes the average values and $\sigma _{\square }$ denotes the root mean square (r.m.s.) value.

Three points (${\rm P}_{\rm S1}$, ${\rm P}_{\rm S2}$ and ${\rm P}_{\rm S3}$) located at the intersection of three slices with the midchord of the wing (see figure 4c,d) have been selected to investigate as the midchord line crosses R1. It is believed that ${\rm P}_{\rm S1}$ and ${\rm P}_{\rm S2}$ are significantly influenced by the evolution of the LEV, while ${\rm P}_{\rm S3}$ is chosen for comparison as it is less influenced by the LEV. Here $R_{\dot {\varGamma }_\tau \dot {p}}$ and $R_{\dot {\hat {\omega }}_i \dot {p}}$ over $t/T=0$ to 0.5 at the three points are summarised in table 1. The sample number $N$ is 50 with an interval of $0.01T$. Here $R_{\dot {\varGamma }_\tau \dot {p}}$ at ${\rm P}_{\rm S1}$ and ${\rm P}_{\rm S2}$ is high (with absolute values larger than 0.8), indicating that the time derivative of surface pressure at these two points is significantly related to the variation of the LEV circulation. Here $R_{\dot {\hat {\omega }}_s \dot {p}}$ and $R_{\dot {\hat {\omega }}_n \dot {p}}$ share a similar value at ${\rm P}_{\rm S1}$ while $R_{\dot {\hat {\omega }}_n \dot {p}}$ ($-$0.79) is more significant than $R_{\dot {\hat {\omega }}_s \dot {p}}$ ($-$0.67) at ${\rm P}_{\rm S2}$. One possible explanation is that the dual stage vortex tilting for the flapping wing (see Chen, Cheng & Wu (Reference Chen, Cheng and Wu2023a); Chen et al. (Reference Chen, Zhou, Werner, Cheng and Wu2023b) for more information) have reoriented the LEV and converted the vorticity from the $\tau$ axis to the $n$ axis. Conversely, a weak cross-correlation is observed at ${\rm P}_{\rm S3}$, suggesting that the pressure variation in the outboard region is less influenced by the LEV, which loses its coherency near the wingtip.

Table 1.

Cross-correlation between $\dot {\hat {\omega }}_s$, $\dot {\hat {\omega }}_n$ and $\dot {\varGamma }_\tau$ at S2 and $\dot {p}$ of ${\rm P}_{\rm S1}$, ${\rm P}_{\rm S2}$ and ${\rm P}_{\rm S3}$.

3.3. Analytical analysis on the vortices and the time derivative of the surface pressure

From the previous section, the evolution of the LEV has been identified as an important factor in determining the time derivative of the surface pressure. Here, the role of the vortices evolution on the time derivative of the surface pressure is further investigated. To do so, an analytical expression of the time derivative of the surface pressure is established with all source terms placed on the right-hand side of the expression. An analysis is conducted to investigate the relationship between the source terms and vortices evolution by incorporating the simulation results. To obtain the expression, the time derivative of the divergence of the momentum equations in conjugation with the incompressible restriction expressed in the body-fixed reference frame $s$–$n$–$\tau$ (Batchelor Reference Batchelor1967; Garmann et al. Reference Garmann, Visbal and Orkwis2013; Jardin & David Reference Jardin and David2015) is as follows:

(3.2)\begin{align} \nabla^2 \frac{\partial p}{\partial t}=\rho\left(2\frac{\partial \hat{Q}}{\partial t} -\underbrace{\frac{\partial [\boldsymbol{\nabla} \boldsymbol{\cdot} (\dot{\boldsymbol{\varOmega}} \times \boldsymbol{r_o})]}{\partial t}}_{\textit{Angular acceleration}} - \underbrace{\frac{\partial [\boldsymbol{\nabla} \boldsymbol{\cdot} ( \boldsymbol{\varOmega} \times (\boldsymbol{\varOmega} \times \boldsymbol{r_o}) )]}{\partial t}}_{\textit{Centrifugal\ acceleration}} -\underbrace{\frac{\partial [\boldsymbol{\nabla} \boldsymbol{\cdot} (2\boldsymbol{\varOmega} \times \boldsymbol{v})]}{\partial t}}_{\textit{Coriolis\ acceleration}} \right), \end{align}

where $\boldsymbol {\varOmega }$ is the rotation rate of the flapping and pitching motions, $\boldsymbol {r_o}$ is the vector from the rotation centre to a given point, $\boldsymbol {v}=\boldsymbol {u}-\boldsymbol {\varOmega } \times \boldsymbol {r_o}$ is the relative velocity and $\hat {Q} =-1/2( \boldsymbol {\nabla } \boldsymbol {v}: \boldsymbol {\nabla } \boldsymbol {v})$ which is analogous to the $Q$-criterion for incompressible flow ($Q=-1/2(\boldsymbol {\nabla } \boldsymbol {u}:\boldsymbol {\nabla } \boldsymbol {u})$, see Jeong & Hussain (Reference Jeong and Hussain1995)).

In (3.2), the divergence of the term $\dot {\boldsymbol {\varOmega }} \times \boldsymbol {r_o}$ is zero. Additionally, it should be noted that the centrifugal acceleration term $-4 \boldsymbol {\varOmega }\boldsymbol {{\cdot }} \dot {\boldsymbol {\varOmega }}$ is a constant over the flow field. As the flapping amplitude and frequency are low, it is negligible compared with the Coriolis acceleration (also reported by Garmann et al. (Reference Garmann, Visbal and Orkwis2013)). Therefore, (3.2) can be simplified as

(3.3)\begin{align} \nabla^2 \frac{\partial p}{\partial t}=\rho\left(2\frac{\partial \hat{Q}}{\partial t} + 4\boldsymbol{\varOmega}\boldsymbol{\cdot} \dot{\boldsymbol{\varOmega}} - \frac{\partial [\boldsymbol{\nabla} \boldsymbol{\cdot} (2\boldsymbol{\varOmega} \times \boldsymbol{v})]}{\partial t}\right)\approx 2\rho\left(\frac{\partial \hat{Q}}{\partial t} - \frac{\partial [\boldsymbol{\nabla} \boldsymbol{\cdot} (\boldsymbol{\varOmega} \times \boldsymbol{v})]}{\partial t}\right). \end{align}

From (3.3), the time derivatives of $\hat {Q}$ and the divergence of the Coriolis acceleration are the major sources of the Poisson equation of $\dot {p}$. Furthermore, Green's second identity is incorporated and yields the following equation:

(3.4)\begin{align} &\iiint G(\boldsymbol{r},\boldsymbol{r}')\nabla^2 \frac{\partial p}{\partial t} - \frac{\partial p}{\partial t}\nabla^2 G(\boldsymbol{r},\boldsymbol{r}') \,{\rm d}V'\nonumber\\ &\quad =\iint G(\boldsymbol{r},\boldsymbol{r}') \frac{\partial }{\partial n}\left(\frac{\partial p}{\partial t}\right) - \frac{\partial p}{\partial t} \frac{\partial G(\boldsymbol{r},\boldsymbol{r}')}{\partial n}\,{\rm d}S', \quad \nabla^2 G(\boldsymbol{r},\boldsymbol{r}')=\delta(\boldsymbol{r}-\boldsymbol{r}'), \end{align}

where $G(\boldsymbol {r},\boldsymbol {r}')$ is Green's function, $\boldsymbol {r}$ is the observer location, $\boldsymbol {r'}$ is the source location, $\delta (\boldsymbol {r}-\boldsymbol {r}')$ is the Dirac function, ${\rm d}V'$ is the flow domain and ${\rm d}S'$ is the wing surface. In the body-fixed reference frame, the two terms on the right-hand side of (3.4), ${\partial }/{\partial n}({\partial p}/{\partial t})$ and $\partial G(\boldsymbol {r},\boldsymbol {r}')/ \partial n$, are almost zero if the volume is properly selected. The solution of (3.3) is approximated as

(3.5)\begin{equation} \frac{\partial p(\boldsymbol{r})}{\partial t} \approx 2\rho \iiint G(\boldsymbol{r},\boldsymbol{r}') \left[ \frac{\partial \hat{Q}(\boldsymbol{r}')}{\partial t} -\frac{\partial [\boldsymbol{\nabla} \boldsymbol{\cdot} (\boldsymbol{\varOmega} \times \boldsymbol{v}(\boldsymbol{r}'))]}{\partial t} \right] {\rm d}V', \end{equation}

which indicates that the surface pressure derivative is related to $\partial \hat {Q}/\partial t$ and $\partial [\boldsymbol {\nabla } \boldsymbol {{\cdot }} (\boldsymbol {\varOmega }\times \boldsymbol {v})]/\partial t$ in the body-fixed reference frame. The two terms on the right-hand side of (3.5) at S1–3 (see figure 4) of $t/T=0.285$ and $t/T=0.42$ are presented in figure 5. It should be noted that S3 is included here as $\dot {p}$ at $t/T=0.285$ near S3 is much larger than that of the inboard region, implying that investigating S3 is physically meaningful.

Figure 5.

Partial derivative $\partial \hat {Q}/\partial t$ ($Q=0$ marked by the black line) at (a) S1, (b) S2 and (c) S3 of $t/T=0.285$; Coriolis acceleration term at (d) S1, (e) S2 and ( f) S3 of $t/T=0.285$; $\partial \hat {Q}/\partial t$ at (g) S1, (h) S2 and (i) S3 of $t/T=0.42$; Coriolis acceleration term at ( j) S1, (k) S2 and (l) S3 of $t/T=0.42$.

At $t/T=0.285$ (see figure 5a–f), the Coriolis acceleration term ($\partial [\boldsymbol {\nabla } \boldsymbol {{\cdot }} (\boldsymbol {\varOmega }\times \boldsymbol {v})]/\partial t$) is much lower with respect to $\partial \hat {Q}/\partial t$ in the three slices. Generally, it can be found that the region with high $\partial \hat {Q}/\partial t$ is mainly enclosed by the boundary of $Q=0$ (marked by the solid black line, considered as the boundary of vortical structures), which indicates that $\partial \hat {Q}/\partial t$ arises from vortices evolution. It can be seen that the LEV grows in size from S1 to S2. Furthermore, $\partial \hat {Q}/\partial t$ in the proximity of the wing surface is negative due to the absorption of the SV (SV2). In contrast, $\partial \hat {Q}/\partial t$ at the outer side of the LEV is positive due to the vorticity fed by the shear layer emanating from the leading edge. The similar distribution pattern of $\partial \hat {Q}/\partial t$ leads to the similar $\dot {p}$ at the wing surface for S1 and S2. At S3, $\partial \hat {Q}/\partial t$ near the wing surface is less prominent but much higher in the wake region where the complex and unsteady vortical interaction (see figure 3d) may account for the higher $\partial \hat {Q}/\partial t$. The different distribution patterns and magnitudes of $\partial \hat {Q}/\partial t$ account for the difference in $\dot {p}$ between S3 and the inner region.

At $t/T=0.42$ (see figure 5g–l), the Coriolis acceleration term is also much less significant than $\partial \hat {Q}/\partial t$, suggesting that $\dot {p}$ is dominated by the latter. Here $\partial \hat {Q}/\partial t$ within the LEV for S1 and S2 is larger than that at $t/T=0.285$ and the LEV begins to detach and moves downstream. The effects of the variation of the distance between the wing surface ($\boldsymbol {r}$) and the source location ($\boldsymbol {r}'$) are represented by changing the value of $G(\boldsymbol {r},\boldsymbol {r}')$ in the integration of (3.5) and lead to the variation of $\dot {p}$, which will be discussed in the next section. At S3, the high-value region of $\partial \hat {Q}/\partial t$ moves upwards compared with that of $t/T=0.285$. Overall, $\partial \hat {Q}/\partial t$ within the LEV of S1 is less than that of S2, which correlates with the lower $\dot {p}$ at the wing surface near S1. Although the high $\partial \hat {Q}/\partial t$ in the wake region of S3 is comparable with that within the LEV of S2, its larger distance from the wing surface renders its influences less significant, which may explain the lower $\dot {p}$ near S3 than that of R1.

3.4. Position of the LEV

Despite the presence of both $\partial \hat {Q}/\partial t$ and the Coriolis acceleration terms, Green's function in (3.4) may play an important role in the formation of R1 and will be elaborated upon here. Green's function $G(\boldsymbol {r},\boldsymbol {r}')$ is related to the distance between the source $\boldsymbol {r}'$ and the wing surface $\boldsymbol {r}$, but its expression cannot be readily given here. In the body-fixed reference frame, $\boldsymbol {r}$ is set as stationary and the only variable in Green's function is $\boldsymbol {r}'$. This indicates that Green's function can be evaluated by reflecting on the source location. From previous section, it is found that $\partial \hat {Q}/\partial t$ within the LEV is the dominant source for R1 (see figure 5h). Therefore, it can be summarised that the effects of Green's function can be reflected by evaluating the location of the LEV (bounded by $Q>0$ and $\omega _{\tau }<0$). In figure 6(a), the LEV with the contours of $Q$ at four specific time steps ($t/T=0.1$, 0.2, 0.42 and 0.49) is shown. Generally, it can be found that the LEV increases its size and moves downward and backward from $t/T = 0.1$ to 0.4, while less variation can be identified between $t/T=0.4$ and 0.49. Furthermore, the location of the LEV should be quantified. Analogous to the concept of the centroid of vorticity (Huang & Green Reference Huang and Green2015; Jones et al. Reference Jones, Medina, Spooner and Mulleners2016), the centroid of $Q$ of the LEV is used to represent the source location and given as follows:

(3.6a,b)\begin{equation} Q_s=\frac{\displaystyle\iint s Q\, {\rm d}A}{\displaystyle\iint Q \,{\rm d}A}, \quad Q_n=\frac{\displaystyle\iint n Q \,{\rm d}A}{\displaystyle\iint Q \,{\rm d}A}, \end{equation}

where $Q_s$ and $Q_n$ are the coordinates of the centroid of the $Q$ within the LEV in the $s$ and $n$ directions, respectively, and $A$ is the region of the LEV (defined in § 3.2).

Figure 6.

Vortex evolution: (a–d) contours of $Q$ within the LEV region at $t/T = 0.1$, 0.2, 0.42 and 0.49, respectively; (e) LEV trajectory at S2 $\tau =0.95c$, represented by the centroid of $Q$ (red arrow denotes the LEV evolution direction); ( f) time histories of the centroid of $Q$ ($Q_s$ and $Q_n$) at S2 from $t/T=0.05$ to 0.5.x.

The $Q$ centroid of the LEV at S2 ($\tau =0.95c$) from $t/T=0.05$ to 0.5 is shown in figure 6(b,c). Note that the leading edge is located at $(s,n) = (0,0)$. From figure 6(a), it can be observed that the LEV generally moves from the leading edge to the trailing edge, detaching from the wing surface to downstream. However, the opposite trend can be found near the end of the half-stroke. The time histories of $Q_s$ and $Q_n$ are shown in figure 6(c) and the extreme values occur near $t/T=0.42$ corresponding to the high $\dot {p}$ for R1, as shown in figure 3. Another interpretation of the formation of R1 is that the detachment of the LEV and the variation of $\hat {Q}$ increase the surface pressure, resulting in a high positive $\dot {p}$ at R1.

3.5. A scaling law for the vortex acoustic source

As discussed in the Section 3.3, $\partial \hat {Q}/ \partial t$ arising from the vortical structures evolution is considered as the main source for $\partial p/\partial t$ which is considered as the dominant source of the far-field noise generated by the hovering wing (see (2.9)). In other words, $\partial \hat {Q}/ \partial t$ can be regarded as the noise source from vortex evolution. It is thus worthy to scale the magnitude of $\partial \hat {Q}/ \partial t$ and $\partial p/\partial t$ with kinematic parameters (e.g. $f_o$) to provide a new insight for MAV design.

According to Lee et al. (Reference Lee, Choi and Kim2015), the vorticity within the LEV of a flapping wing is estimated by

(3.7)\begin{equation} {\omega} \sim \frac{U_{ref} c AR \sin(\alpha)}{A_{vor}(AR + 2)}, \end{equation}

where $A_{vor}$ is the cross area of the vortex, and $\alpha$ is the angle of attack during the middle stroke. Therefore, $\hat {Q}$ can be estimated as

(3.8)\begin{equation} \hat{Q} \simeq \frac{1}{4} \omega^2 \sim \frac{U_{ref}^2 c^2 AR^2 \sin^2(\alpha)}{4A_{vor}^2(AR + 2)^2} = \frac{4{\rm \pi}^2 c^4 AR^2(AR+0.1)^2 \sin^2(\alpha)}{25 A_{vor}^2(AR + 2)^2}f_0^2, \end{equation}

where $U_{ref} = 4{\rm \pi} (AR+0.1)c f_0/5$ is applied. The change of $\hat {Q}$ happens within a period of flapping, and thus

(3.9)\begin{equation} \frac{\partial \hat{Q}}{\partial t} \sim \frac{4{\rm \pi}^2 c^4 AR^2(AR+0.1)^2 \sin^2(\alpha)}{25 A_{vor}^2(AR + 2)^2}f_0^3. \end{equation}

The scaling law for the pressure variation with respect to time is achieved by applying (3.9) into (3.5), i.e.

(3.10)\begin{equation} \frac{\partial p}{\partial t} \sim A_{vor}\partial \hat{Q}/\partial t \sim \frac{4{\rm \pi}^2 c^4 AR^2(AR+0.1)^2 \sin^2(\alpha)}{25 A_{vor}(AR + 2)^2}f_0^3. \end{equation}

The r.m.s. values of $\dot {f_{p_i}}$ (see (2.8)) are incorporated here as an indicator to show whether the $f_o^3$ law is valid. Here, five different flapping frequencies are chosen as $f_o= \{0.05,0.1,0.125,0.2,0.25\}$, yielding five different Reynolds numbers ranging from 200 to 1000. Here $\dot {f_{p_i}}$ is calculated after twenty flapping cycles where the cycle-to-cycle variations in the pressure are negligible. The second-order central difference is adopted to calculate the time derivative of the surface pressure. The r.m.s. values of $\dot {f_{p_i}}$ are evaluated over five flapping cycles as shown in figure 7, where the r.m.s. of $\dot {f_{p_i}}$ is proportional to $f_o^3$ for all three directions. The coefficients in (3.9) and (3.10) appear as a minor shift in the vertical direction, and thus are not explicitly included. Therefore, the vortex acoustic source $\partial \hat {Q}/\partial t$ is scaled by $f_o^3$. It should be noted that the scaling law, (3.10), is a very useful tool in MAV design and other engineering applications.

Figure 7.

Scaling law for the time derivatives of pressure forces.