3.1. Effect of gap width on aerodynamic forces

Figure 4.

Effect of gap width on average nominal aerodynamic force coefficients. (a) Average nominal lift coefficient $\overline {{C}_{L,n}}$ . (b) Average nominal lift-to-drag ratio ${\overline {{C}_{L,n}}}/{\overline {{C}_{D,n}}}$ . The solid line denotes the results of the present unsteady simulation, the dashed line corresponds to the results of the quasi-2-D model, where the aerodynamic force is proportional to the lifting-surface area and the dash–dotted line reflects the results of the quasi-steady lift model of Nabawy & Crowther (Reference Nabawy and Crowther2014b ). The case of $w=0$ corresponds to the intact wing.

Figure 4 illustrates the effect of gap width on the average nominal aerodynamic force coefficients. The solid line with triangular symbols represents the results of the present numerical simulation. The moult-gapped wing with a normalised gap width $w \leqslant 0.1$ almost maintains the average nominal lift coefficient $\overline {{C}_{L,n}}$ , which is equivalent to that of the intact wing. As the normalised gap width $w$ increases to 0.2 and 0.3, the average nominal lift coefficient $\overline {{C}_{L,n}}$ of the moult-gapped wing decreases to 0.71 and 0.63, respectively. To identify the factors that affect the aerodynamics of the moult-gapped wing, we introduce the results based on two theoretical models in figure 4. The dashed line with circular symbols is derived from the quasi-two-dimensional (quasi-2-D) model, which assumes that the aerodynamic force is proportional to the lifting-surface area. The cases denoted by the dashed line employ the same aerodynamic forces per unit area as those of the intact wing. In addition, the dash–dotted line with square symbols in figure 4(a) represents the predicted results of the quasi-steady lift model for general intact flapping wings. This quasi-steady lift model is proposed by Nabawy & Crowther (Reference Nabawy and Crowther2014b ), and includes both translational and non-translational components. The non-translational components are equal to zero, since the flapping wing in this work does not rotate about its spanwise axis during stroke reversal, and the elevation motion is symmetric between the upstroke and downstroke. The translational component dominates the generation of average lift and is expressed as the function of the geometric angle of attack $\alpha$ as

(3.1) \begin{equation} (\overline {C_{L,a}})^{ \textit{quasi-st.}}=\frac {{{{\kappa }_{ {2D}}}}/{2}} {E+{{{\kappa }_{ {2D}}}}/({\pi {e}{{AR}_{a}}})}\sin \left ( 2\alpha \right ), \end{equation}

where $\kappa _{ {2D}}$ is the lift-curve slope of a two-dimensional flat-plate aerofoil, which is set to $5.16$ to account for the effect of the Reynolds number (Okamoto, Yasuda & Azuma Reference Okamoto, Yasuda and Azuma1996). The geometry-specific parameter $E$ is the Jones edge velocity correction for the variation in the local chord of finite span wing (Jones Reference Jones1941). The Jones edge velocity correction is defined as the ratio of the wing semi-perimeter to the wing span, which is $3/2$ and $11/6$ for the intact and moult-gapped wings, respectively. Here, ${AR}_{a}$ is the actual aspect ratio of a single wing, which is given by ${{b}^2}/4{S_a}$ . The span efficiency factor $e$ corresponds to the induced angle of attack due to the finite span (Dommasch, Sherby & Connolly Reference Dommasch, Sherby and Connolly1961). This span efficiency factor is 0.75, which is close to the mean value evaluated by Muijres et al. (Reference Muijres, Spedding, Winter and Hedenström2011, Reference Muijres, Bowlin, Johansson and Hedenström2012) and Nabawy & Crowther (Reference Nabawy and Crowther2014a ) for flapping flight. The nominal lift coefficient predictions of this formula are obtained through a reference area conversion.

The intact wing is represented in figure 4 by a normalised gap width $w=0$ to clarify the effect of the moult gap. The average nominal lift coefficient $\overline {{C}_{L,n}}$ (a measure of the total aerodynamic force) of the intact wing is 0.76, which agrees well with that of the quasi-steady lift model. The average nominal lift coefficient $\overline {{C}_{L,n}}$ of the moult-gapped wing does not exceed that of the intact wing owing to the loss of the lifting surface. For these moult-gapped wings, the average nominal lift coefficient $\overline {{C}_{L,n}}$ almost decreases with an increase in the gap width.

The average nominal lift coefficient has the slowest decline rate relative to the normalised gap width when compared with the results from the quasi-2-D and quasi-steady lift models. The quasi-2-D lift model exhibits a linear decrease induced solely by the reduction in the lifting-surface area, whereas the quasi-steady lift model presents a more rapid decrease owing to the finite wing span. This comparison suggests that the loss of the lifting surface not only reduces the area supporting the aerodynamic loading, but also triggers a nonlinear flow mechanism. The nonlinear flow mechanism mitigates the reduction in aerodynamic force caused by the moult gap.

Despite its benefits, this nonlinear flow mechanism cannot prevent a decrease in the average lift-to-drag ratio of the moult-gapped wings, as shown in figure 4(b). The average nominal lift-to-drag ratio ${\overline {{C}_{L,n}}}/{\overline {{C}_{D,n}}}$ of the intact wing is 1.45. The moult-gapped wing with a normalised gap width of $w=0.1$ has an average nominal lift-to-drag ratio ${\overline {{C}_{L,n}}}/{\overline {{C}_{D,n}}}$ of 1.41, which is 3 % smaller than that of the intact wing. This results from an increase in the average nominal drag coefficient $\overline {{C}_{D,n}}$ that occurs during the early stages of moulting in the investigation of gliding flight (KleinHeerenbrink & Hedenström Reference KleinHeerenbrink and Hedenström2017). As the gap width increases, the effect of the reduction in the lifting-surface area is revealed. When the normalised gap width reaches 0.15, the average nominal lift-to-drag ratio ${\overline {{C}_{L,n}}}/{\overline {{C}_{D,n}}}$ is close to the value of the intact wing owing to the almost identical average nominal drag coefficient $\overline {{C}_{D,n}}$ . This almost-identity results from the cancellation between the reduction in the lifting-surface area and the increase in the drag per unit area. For the wide gaps ( $w=0.2$ and $w=0.3$ ), the decrease in the average nominal lift-to-drag ratio ${\overline {{C}_{L,n}}}/{\overline {{C}_{D,n}}}$ is attributed to the larger decrease in the average nominal lift coefficient $\overline {{C}_{L,n}}$ compared with that in the average nominal drag coefficient $\overline {{C}_{D,n}}$ . As observed by Sunada et al. (Reference Sunada, Takashima, Hattori, Yasuda and Kawachi2002) and Lee & Kim (Reference Lee and Kim2017), the average nominal lift coefficient $\overline {{C}_{L,n}}$ decreases more rapidly than the average nominal drag coefficient $\overline {{C}_{D,n}}$ under wide gaps, which Lee & Kim (Reference Lee and Kim2017) ascribed to the higher sensitivity of lift compared with drag to changes in the wing shape.

Figure 5.

Effect of gap width on average actual lift coefficient $\overline {{C}_{L,a}}$ . The legends are the same as those in figure 4. The case of $w=0$ corresponds to the intact wing.

The actual aerodynamic coefficient reflects the relative magnitude of the decrease in the aerodynamic force and reduction in area. Figure 5 shows the average actual lift coefficients of the intact and moult-gapped wings. Because the proportion of lifting-surface reduction is larger than that of aerodynamic lift reduction, the average actual lift coefficients of the moult-gapped wing are greater than those of the intact wing. Larger average actual aerodynamic force coefficients of the moult-gapped wing were also observed in experimental measurements of hovering hummingbirds by Chai (Reference Chai1997). Among these moult-gapped wings, those with normalised gap widths of 0.1 and 0.2 exhibit equal reduction proportions in average lift and area, resulting in an equal average actual lift coefficient of 0.82. When the normalised gap width increases to 0.3, the average actual lift coefficient decreases to 0.79. The average actual lift-to-drag ratio ${\overline {{C}_{L,a}}}/{\overline {{C}_{D,a}}}$ exactly equals the average nominal lift-to-drag ratio ${\overline {{C}_{L,n}}}/{\overline {{C}_{D,n}}}$ because the difference between them lies only in the reference area. The quasi-steady lift model cannot accurately detect unsteady interactions caused by moult gaps, resulting in an underestimation of the average actual lift coefficient. Table 3 shows the difference in the average actual lift coefficient between the results of the present unsteady simulation and quasi-steady lift model. The quasi-steady lift model underestimates by up to 16 % compared with the numerical simulation when the gap width is 0.1. Even for the moult-gapped wing with the gap width of 0.3, which has the smallest underestimation by the model, the difference in average actual lift coefficient is still 8 %.

Table 3.

Difference in the average actual lift coefficient between the unsteady simulation and quasi-steady model of Nabawy & Crowther (Reference Nabawy and Crowther2014b ).

To investigate the change in the average nominal aerodynamic force coefficients, the histories of the instantaneous nominal aerodynamic coefficients of the moult-gapped wings with different gap widths during the flapping period are plotted in figure 6. The white and shaded blocks represent the upstroke and downstroke of the flapping wing, respectively. When the normalised gap width $w$ is 0.1, during the downstroke, the curves of the nominal lift coefficient ${C}_{L,n}$ and nominal drag coefficient ${C}_{D,n}$ of the moult-gapped wing are slightly different from those of the intact wing. This slight difference shows that the moult-gapped wing with a normalised gap width $w$ of 0.1 does not suffer a significant loss in aerodynamic force owing to the lifting surface reduction. As the gap width increases (the normalised gap width $w$ increases to 0.2 and 0.3), the instantaneous lift and drag gradually decrease. As shown in figure 6, the maximum decrease in the aerodynamic force owing to the lifting-surface loss occurs near the peak of aerodynamic force during the downstroke. Near the moment of maximum aerodynamic force, the decrease in the force is faster than the decrease in the area for wide gaps. At the moment of $t/T=3.45$ , for normalised gap widths $w$ of 0, 0.1, 0.2 and 0.3, the ratio of the nominal lift coefficients is $1:0.96:0.85:0.73$ , whereas the ratio of the actual lifting-surface areas is $1:0.93:0.87:0.80$ . Conversely, the further away from the moment of maximum aerodynamic force, the weaker the effect of area loss. During the upstroke, the aerodynamic forces of the wings with different gap widths tend to be consistent. The moult gap enhances the aerodynamic force relative to the intact wing during part of the upstroke (see the time intervals $3.00\lt t/T\lt 3.10$ and $3.90\lt t/T\lt 4.00$ in figure 6).

Figure 6.

Effect of the gap width on the total aerodynamic forces of the moult-gapped wing. (a) Nominal lift coefficient ${C}_{L,n}$ . (b) Nominal drag coefficient ${C}_{D,n}$ . The subscript $n$ indicates that the reference area is the area of the intact wing. The white and shaded blocks represent the upstroke and downstroke of the flapping wing, respectively.

The larger average actual lift coefficient of the moult-gapped wing compared with the intact one is a consequence of temporal accumulation, as shown in figure 7. To facilitate comparison, the differences in the aerodynamic coefficients between the moult-gapped and intact wings are also plotted. The positive increment in the instantaneous actual lift coefficient ${C}_{L,a}$ of the moult-gapped wing persists for most of the flapping period, although the positive increment is small. The moult-gapped wing only exhibits a lower actual lift coefficient ${C}_{L,a}$ during the time interval $3.70 \lt t/T \lt 3.86$ when the normalised gap width $w$ is 0.1, and during the acceleration stage $3.25 \lt t/T \lt 3.50$ of the downstroke when the normalised gap width $w$ is 0.2 and 0.3. The actual lift coefficient curve of the moult-gapped wing, with a normalised gap width of 0.3, decreases more near the peak. It is worth noting that despite the different gap sizes, the actual drag coefficient ${C}_{D,a}$ of the moult-gapped wing is always greater than that of the intact wing. During most of the upstroke, the actual drag coefficient ${C}_{D,a}$ increases monotonically as the gap widens. When these results are considered alongside the experimental data from KleinHeerenbrink & Hedenström (Reference KleinHeerenbrink and Hedenström2017), it is plausible that increased drag is a common feature that is consistent in both gliding and flapping flight modes.

Figure 7.

Effect of the gap width on the aerodynamic forces per unit area of the moult-gapped wing. (a) Actual lift coefficient ${C}_{L,a}$ . (b) Actual drag coefficient ${C}_{D,a}$ . (c) Difference in the actual lift coefficient ${\Delta }{C}_{L,a}$ . (d) Difference in the actual drag coefficient ${\Delta }{C}_{D,a}$ . The subscript $a$ indicates that the reference area is the area of the actual wing. The white and shaded blocks represent the upstroke and downstroke of the flapping wing, respectively.

3.2. Vortex dynamics of moult-gapped wings

The above-mentioned analysis of the aerodynamic forces of moult-gapped wings with different gap widths shows that the moult-gapped wing can generate more aerodynamic force per unit area. To reveal the vortex dynamics associated with these changes, the vortices of the intact and moult-gapped wings are compared in this subsection.

Figure 8 shows the vortices of the intact wing and the moult-gapped wing with a normalised gap width $w$ of 0.2 at three equally spaced moments during the upstroke deceleration stage ( $3.00\lt t/T\lt 3.25$ ). These vortices are identified using isosurfaces of the $Q$ -criterion (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988) and coloured by the spanwise vorticity. For the intact wing, the LEVs appear along the entire span of the wing with their streamwise size increasing continuously, as the flapping velocity of the moult-gapped wing gradually decreases to zero, as shown in figure 8(a). Simultaneously, the TEVs are convected towards the wake. The TVs swing from the lower side to the upper side relative to the wing plane.

The flow over the moult-gapped wing contains vortices with similar characteristics. However, the moult gap at the trailing edge induces distinct local vortices. The most important feature is that the shear layer at the edges of the moult gap develops into a type of attached vortex structures. We refer to these vortex structures as gap vortices (GVs), according to the work of Lee, Lahooti & Kim (Reference Lee, Lahooti and Kim2018). To illustrate the features of GVs, figure 9(a) presents an isosurface with the $Q$ -criterion of 5 to depict the three-dimensional structure of GVs around the moult-gapped wing with a normalised gap width $w$ of 0.2. The GVs are composed of three segments attached to the inner, leading and outer edges of the moult gap. The GVs use their inner and outer edge segments to connect the TEVs of the moult-gapped wing. As shown in figure 9(b), the inner and outer edge segments are a pair of counter-rotating streamwise vortices with comparative strengths. The magnitude of this pair of vortices is larger at the mid-chord position of the moult gap than at both ends.

Figure 8.

Vortices in the flow over the intact and moult-gapped wings during the upstroke deceleration stage ( $3.00\lt t/T\lt 3.25$ ). (a) Intact wing. (b) Moult-gapped wing (the normalised gap width ${w}=0.2$ ). Free stream flow is directed from left to right. The vortex structures are represented by isosurfaces of $Q=3$ and coloured by the normalised spanwise vorticity ${\omega }_{y}$ .

Figure 9.

Gap vortices (GVs) around the moult-gapped wing. (a) Spacial structure. (b) Normalised streamwise vorticity in the inner ( $\triangle$ ) and outer ( $\square$ ) edge segments. The plot shows the attached vortex structures of the moult-gapped wing with a normalised gap width $w$ of 0.2 at the moment $t/T=3.30$ . The vortex structures are represented by isosurfaces of $Q=5$ in panel (a). The normalised streamwise vorticity ${\omega }_{x}$ is displayed in four slices at normalised streamwise positions of –0.1, 0, 0.1 and 0.2 in panel (b).

Figure 10.

Vortices in the flow over the intact and moult-gapped wings during the downstroke. (a) Intact wing. (b) Moult-gapped wing (the normalised gap width ${w}=0.2$ ). Free stream flow is directed from left to right. The vortex structures are represented by isosurfaces of $Q=3$ and coloured by the normalised spanwise vorticity ${\omega }_{y}$ .

Figure 11.

Normalised spanwise vorticity in the flow over the intact and moult-gapped wings during the downstroke. (a) Intact wing. (b) Moult-gapped wing (the normalised gap width ${w}=0.2$ ). Free stream flow is directed from left to right. The spanwise distance from the slice to the symmetry plane at the wing root is 56 % of the semi-span. The slice crosses the narrow section with the moult gap.

When the intact wing and the moult-gapped wing with a normalised gap width $w$ of 0.2 enter the downstroke, the respective vortex structures at four equally spaced moments (the moment ${t}/{T}=3.76$ represents the end of the downstroke) are shown in figure 10. To elucidate the effects of the moult gap, figure 11 presents the distribution of spanwise vorticity ${\omega }_{y}$ in a slice located 56 % of the semi-span from the symmetry plane at the wing root. The slice passes through narrow sections of the moult-gapped wing with a normalised gap width $w$ of 0.2 throughout the downstroke. Before the mid-downstroke ${t}/{T}=3.50$ , the streamwise size of the LEVs over the intact wing continues to increase and gradually evolves to a spiral configuration, as observed in figures 10(a) and 11(a). The TEVs of the intact wing are convected downstream during the downstroke acceleration stage ( $3.25\lt t/T\lt 3.50$ ). The TEVs shed in the form of a single semi-ring together with the TVs at the moment ${t}/{T}=3.63$ in the following deceleration stage.

For the moult-gapped wing, the streamwise development of the LEVs produces their interaction with the GVs, resulting in a reduction in the streamwise size of LEVs compared with that of the intact wing (at the moment ${t}/{T}=3.37$ ). The interaction between the LEVs and GVs also changes the spanwise distribution and shedding process of the LEVs, as shown in the three windows in figure 10(b). Furthermore, the moult gap changes the evolution of the TEVs, leading to the TEVs shedding in the form of double semi-rings at the moment ${t}/{T}=3.63$ . This is because of the simultaneous shedding of the inner and outer edge segments of the GVs and their connected TEVs.

The flow at the moult gap is characterised by the appearance of the GVs. The evolution of GVs remains consistent as the gap width varies. The GVs are responsible for aerodynamic compensation for the lifting-surface loss of the moult-gapped wings. The GVs induce nonlinear interactions among the vortices generated in the flapping flight. The GVs affect the evolution of other main vortex structures, thereby altering the vortex lift. At the same time, the GVs adjust the gap leakage flow, which modifies the near wake by changing the flow momentum (detailed in § 3.3). The moult-gapped wing relies on these interactions to compensate for the lost aerodynamic force, which is identified as vortex compensation. This vortex compensation corrects the discrepancy between the results ( $\overline {C_{L,a}}$ ) of the unsteady simulation and the predictions ( $(\overline {C_{L,a}})^{ {quasi\text{-}st.}}$ ) of the quasi-steady lift model (3.1) by

(3.2) \begin{equation} \overline {C_{L,a}}=(\overline {C_{L,a}})^{ \textit{quasi-st.}}+(\overline {C_{L,a}})^{ {\textit{gap}}}, \end{equation}

where $(\overline {C_{L,a}})^{ {\textit{gap}}}$ is the contribution of the vortex compensation.

The vortices around the flapping wing can be modelled as a vortex loop (Hedenström et al. Reference Hedenstr̈m, Johansson, Wolf, von Busse, Winter and Spedding2007). The change in the vortex loop from the intact wing to the moult-gapped wing is displayed in figure 12. This change affects unsteady lift, thereby modifying average lift (Wang & Wu Reference Wang and Wu2010; Chowdhury & Ringuette Reference Chowdhury and Ringuette2019). The correlation between the vortex loops and the average lift can be described by an actuator disk model (Spalart Reference Spalart2003; Oh et al. Reference Oh, Lee, Park, Choi and Kim2020). To compute the average lift of flapping wings in forward flight, Muijres et al. (Reference Muijres, Spedding, Winter and Hedenström2011) have modified the actuator disk model, which involves the unsteady vortex dynamics by using the non-uniform vertical induced velocity and the modified actuator disk area. The modified actuator disk area is selected as the area enclosed by the main vortex structures developed within one flapping period. The average lift is the change rate of downward momentum across the control surface around the flapping wing. The downwash flow at the actuator disk is visualised in figure 12(a).

Figure 12.

Schematic of the vortex loops formed by the main vortex structures. (a) Intact wing. (b) Moult-gapped wing. The dashed lines represent the vortex lines, with thin arrows indicating their directions. The bold arrows denote the downwash flow at the modelled actuator disk. An observation slice is visualised in panel (b).

Figure 13.

Generation of the increased induced downwash at the double vortex loops. The schematic is plotted on the observation slice in figure 12. The upward gap leakage through the moult gap creates the GVs, which increase the induced downwash. The variables $r_{ {\textit{in}}}$ and $r_{ {\textit{out}}}$ are the radial positions of the inner and outer edge segments of the GVs, respectively.

Figure 14.

Normalised vertical velocity $u_z$ . (a) Intact wing. (b) Moult-gapped wing (the normalised gap width ${w}=0.2$ ). The slice A is at the spanwise position $2y/b=78\,\%$ , and the slice B is at the streamwise position $2x/b=45\, \%$ . The plot shows the downwash behind the intact wing and its increase behind the moult-gapped wing at the moment $t/T=3.63$ .

We developed a new model to account for the interaction caused by the moult gap. The moult gap with the high ratio of the longitudinal length to transverse width splits the original single vortex loop into double vortex loops, as illustrated in figure 12(b). We model the vortex compensation associated with the moult gap as an increase in vertical induced velocity at the actuator disks corresponding to the double vortex loops, as shown in figure 13. The increase in the induced downwash results from the GVs, which are generated by the upward gap leakage through the moult gap. Figure 14 exhibits the induced downwash behind the intact wing and its increase behind the moult-gapped wing during the downstroke. The intact wing creates more downwash behind its outer part owing to a higher relative velocity. The moult-gapped wing increases the induced downwash more prominently in the outer vortex loop. The upward gap leakage vanishes in the surrounding downwash after travel of approximately a semi-chord. In our model, the increased induced downwash correlates with the actual lifting surface, whereas the upward gap leakage corresponds to the lost lifting surface. The average lift component compensating for the lifting-surface loss is represented by the change rate of upward momentum. The normalised form of the average lift component is given as follows:

(3.3) \begin{equation} (\overline {C_{L,a}})^{ {\textit{gap}}}=\frac {1}{{1}/{2}\;\rho U_{\infty }^{2}{{S}_{a}}}\int _{{{r}_{ {\textit{in}}}}}^{{{r}_{ {\textit{out}}}}}{\rho {{\left [ {{u}_{i}}\left ( r \right ) \right ]}^{2}}\frac {2c}{3}}\,{\rm d}r=\frac {4c}{3U_{\infty }^{2}{{S}_{a}}}\int _{{{r}_{ {\textit{in}}}}}^{{{r}_{ {\textit{out}}}}}{{{\left [ {{u}_{i}}\left ( r \right ) \right ]}^{2}}}\,{\rm d}r, \end{equation}

where $r$ is the radial distance from the wing root, ${{u}_{i}}(r)$ is the induced upwash velocity of the GVs within the moult gap. Here, ${r}_{ {\textit{in}}}$ and ${r}_{ {\textit{out}}}$ are the radial positions of the inner and outer edges of the moult gap, respectively, that is,

(3.4) \begin{equation} {{r}_{ {\textit{in}}}}=\frac {b}{4}+d\frac {b}{2}, \end{equation}

(3.5) \begin{equation} {{r}_{ {\textit{out}}}}=\frac {b}{4}+(d+w)\frac {b}{2}. \end{equation}

The evolution of the GVs with wing motion indicates that the strength of the GVs depends on the local relative velocity between the wing and fluid. This relative velocity is dominated by the local elevation velocity under a specified incoming velocity. The induced upwash velocity is proportional to the vertical component of local elevation velocity as

(3.6) \begin{equation} {{u}_{i}}\left ( r \right )\propto {{u}_{e,z}}\left ( r \right ). \end{equation}

Here,

(3.7) \begin{equation} {{u}_{e,z}}\left ( r \right )=\left ( {{\varOmega }}r \right )\cos \alpha , \end{equation}

where $\varOmega$ is the effective elevation velocity, which is defined as the root-mean-square of the elevation velocity, that is, the maximum elevation velocity multiplied by a factor of ${\sqrt {2}}/{2}$ ,

(3.8) \begin{equation} {{\varOmega }}=\frac {\sqrt {2}}{2}\left ( 2\pi f{{\psi }_{m}} \right )=\sqrt {2}\pi f{{\psi }_{m}}. \end{equation}

The effective elevation velocity reflects the average energy of the moult-gapped wing during an elevating period (Sujatha Reference Sujatha2010; Wang et al. Reference Wang, Xu, Zhang and Wang2021). This characteristic quantity accounts for the cumulative effect of the elevating velocity at all moments, reflecting the complete temporal behaviour of the elevating motion. The effective elevation velocity also matches the average lift component in (3.3), which is formulated as the integral of the squared induced velocity.

From the relation (3.6), we define a correlation coefficient $k$ as the ratio of the induced upwash velocity to the vertical component of local elevation velocity for measuring their correlation,

(3.9) \begin{equation} k=\frac {{{u}_{i}}\left ( r \right )}{{{u}_{e,z}}\left ( r \right )}. \end{equation}

Substituting (2.6) and (3.4)–(3.9) into (3.3), and rearranging the resulted expression, we obtain the compact form

(3.10) \begin{equation} (\overline {C_{L,a}})^{ {\textit{gap}}}=\frac {16}{9} G {{k}^{2}} {{\left ( \frac {\varOmega b}{2{{U}_{\infty }}} \right )}^{2}} {{\cos }^{2}}\alpha , \end{equation}

where the average lift coefficient accounting for the moult gap is written in terms of the correlation coefficient, inverse of advance ratio ( ${2{{U}_{\infty }}}/{\varOmega b}$ ), geometric angle of attack, and a geometric parameter related to the width and position of the moult gap,

(3.11) \begin{equation} G = \frac {\displaystyle \left [ {{\left ( \frac {1}{4}+\frac {d+w}{2} \right )}^{3}}-{{\left ( \frac {1}{4}+\frac {d}{2} \right )}^{3}} \right ]}{\displaystyle \left ( \frac {1}{2}-\frac {w}{3} \right )}. \end{equation}

According to (3.1) and (3.10), the final expression for the average actual lift coefficient is

(3.12) \begin{equation} \overline {C_{L,a}} =\frac {{{{\kappa }_{ {2D}}}}/{2}} {E+{{{\kappa }_{ {2D}}}}/({\pi {e}{{AR}_{a}}})}\sin \left ( 2\alpha \right ) + \frac {16}{9} G {{k}^{2}} {{\left ( \frac {\varOmega b}{2{{U}_{\infty }}} \right )}^{2}} {{\cos }^{2}}\alpha . \end{equation}

The correlation coefficient $k$ depends on both the width and the position of the moult gap. To decouple the effects of these two factors, $k$ is decomposed into two parts as follows:

(3.13) \begin{equation} k={{k}_{w}}{{k}_{d}}, \end{equation}

where $k_w$ and $k_d$ account for the effects of the gap width and gap position, respectively. We make a convention in this section that when the gap is located a wing-mid position of $d = 0$ , the correlation coefficient $k_d = 1$ . The correlation coefficient $k$ can be solved by combining (3.1)–(3.13). As shown in figure 15(a), the correlation coefficient $k_w$ decreases as the normalised gap width $w$ increases. The fitted linear model for the correlation coefficient $k_w$ is

(3.14) \begin{equation} k_w = -17.9w + 6.4. \end{equation}

The decrease trend implies that a wider moult gap cannot effectively induce an upwash flow through the GVs owing to an increased distance from the gap flow to the vortex cores of the GVs, thereby weakening the vortex compensation. Using (3.14), the newly proposed lift model (3.12) can reasonably predict the average actual lift coefficient of moult-gapped wings with different gap widths, as illustrated in figure 16(a).

Figure 15.

Correlation coefficients (a) $k_w$ and (b) $k_d$ between the induced upwash velocity and vertical component of local elevation velocity for moult-gapped wings with different gap widths and positions. The diamond symbols represent the results of numerical simulations, while the solid lines correspond to the fitted models.

Figure 16.

Comparison of the proposed model prediction with numerical simulation for average actual lift coefficient. (a) Moult-gapped wings with different gap widths. (b) Moult-gapped wings with different gap positions. The triangle symbols represent the results of numerical simulations, while the solid lines correspond to the proposed lift models.

3.3. Moult gaps at different spanwise positions

Figure 17.

Effect of gap position on average nominal aerodynamic force coefficients. (a) Schematic of moult gaps at different spanwise positions. (b) Average nominal lift coefficient $\overline {{C}_{L,n}}$ . (c) Average nominal drag coefficient $\overline {{C}_{D,n}}$ . The dashed line in panels (b) and (c) represents the average nominal aerodynamic force coefficients of the intact wing.

To clarify the vortex compensation generated by the moult gaps at different spanwise positions, we present the results for a constant normalised gap width ${w}=0.1$ and distinct normalised gap positions $d$ , as detailed now. As displayed in figure 17(a), when the normalised gap position $d=0$ , the inner edge of the moult gap is located in the wing mid. When the normalised gap position $d=0.4$ , the outer edge of the moult gap is located at the wing tip, whereas when the normalised gap position $d=-0.4$ , the distance between the inner edge of the moult gap and the wing root is $b/20$ . The effect of gap position on the average nominal aerodynamic force coefficients is shown in figures 17(b) and 17(c). To highlight the aerodynamics of the moult-gapped wings, we use dashed lines to indicate the average nominal lift coefficient $\overline {{{C}_{L,n}}}=0.76$ and the average nominal drag coefficient $\overline {{{C}_{D,n}}}=0.52$ of the intact wing.

For the cases investigated in this section, when the moult gap is located in the inner part of the moult-gapped wing (the normalised gap position $d\lt 0$ ), the average nominal lift coefficient $\overline {{{C}_{L,n}}}$ and average nominal drag coefficient $\overline {{{C}_{D,n}}}$ of the moult-gapped wing are not lower than the corresponding values of the intact wing. Hence, moult-gapped wings with inboard moult gaps tend to maintain or even enhance aerodynamic forces. If the normalised gap position $d$ is −0.1, the average nominal lift coefficient $\overline {{{C}_{L,n}}}$ is almost equal to that of the intact wing. As the moult gap moves inwards, the average nominal lift coefficient of the moult-gapped wing increases. When the normalised gap positions are −0.2 and −0.3, the average nominal lift coefficients are approximately 0.78. When the moult gap moves further inwards to where the normalised gap position $d$ is −0.4, the average nominal lift coefficient of the moult-gapped wing reaches 0.82, an 8 % increase from that of the intact wing. The average nominal drag coefficient of the moult-gapped wing varies relatively gently with changes in the gap position along the span. As the normalised gap position shifts from −0.1 to −0.3, the average nominal drag coefficient of the moult-gapped wing almost remains constant at 0.54. Until the normalised gap position reaches −0.4 near the wing root, the average nominal drag coefficient of the moult-gapped wing experiences a maximum increase of 6 %.

When the moult gap is located in the outer part of the moult-gapped wing (the normalised gap position $d \geqslant 0$ ), the average nominal lift coefficient $\overline {{C}_{L,n}}$ may be lower than the corresponding values for the intact wing. Moult-gapped wings with outboard moult gaps could lose partial aerodynamic lift. For example, when the gap location parameter $d$ is 0.1, the average nominal lift coefficient $\overline {{C}_{L,n}}$ of the moult-gapped wing decreases to 0.74. As the moult gap relocates outwards to the position corresponding to a normalised gap position $d$ of 0.2, the average nominal lift coefficient of the moult-gapped wing reverts to the baseline level observed for the intact wing. With the normalised gap position $d$ of 0.3, there is a 3 % increase in the average nominal lift coefficient $\overline {{C}_{L,n}}$ . Upon further outwards displacement to the wing tip, where the normalised gap position $d$ reaches 0.4, the average nominal lift coefficient $\overline {{C}_{L,n}}$ of the moult-gapped wing decreases to 0.75. The average nominal drag coefficient of the moult-gapped wing is not lower than that of the intact wing. The outboard moult gaps yield a smaller average nominal drag coefficient than the inboard ones of the wing.

The quasi-steady lift model cannot predict the change in the average actual lift coefficient caused by the different spanwise positions of the moult gap because all parameters in (3.1) have the same values. However, the effect of gap position is considered in the moult-gap-related geometric quantity and the correlation coefficient $k_d$ of the proposed model (3.10). As shown in figure 15(b), the correlation coefficient $k_d$ decreases in a negative power law as the moult gap moves outwards, eliminating the irregularity of the change in the average lift coefficient. The fitted power law is

(3.15) \begin{equation} k_d = \frac {1}{1.9d+1}. \end{equation}

The small correlation coefficient near the wing tip could result from the characteristics of LEVs and TVs, as well as the gap flow induced by high elevation velocity (detailed later). With (3.15), the newly proposed lift model (3.12) qualitatively predicts the trend of the average actual lift coefficient with respect to the spanwise position of the moult gap, as shown in figure 16(b).

The effect of gap position on the aerodynamic performance of a finite-span wing is more complex during flapping flight than during gliding flight. In gliding flight, the aerodynamic performance of the moult-gapped wing progressively enhances as the moult gap relocates outwards owing to the reduction in the spanwise circulation within the cross-section towards the wing tip (Hedenström & Sunada Reference Hedenström and Sunada1999). Conversely, in flapping flight, despite a comparable decline in the instantaneous and time-averaged spanwise circulations from the proximal end to the distal end along the span (Poelma, Dickson & Dickinson Reference Poelma, Dickson and Dickinson2006; Thielicke & Stamhuis Reference Thielicke and Stamhuis2015), the aforementioned trend of the aerodynamic force of the moult-gapped wing markedly differs. To further elucidate the effect of the gap position on the aerodynamic forces of the moult-gapped wing, we use the intact wing as a reference and examine the instantaneous force coefficients over one flapping period at two gap positions ( $d=0.3$ and $d=-0.4$ ) related to the average aerodynamic force enhancement and two gap positions ( $d=0.1$ and $d=0.4$ ) associated with the average aerodynamic force loss.

Figure 18.

Instantaneous nominal aerodynamic force coefficients of the moult-gapped wings that enhance the aerodynamic forces. (a) Nominal lift coefficient ${C}_{L,n}$ . (b) Nominal drag coefficient ${C}_{D,n}$ . The white and shaded blocks represent flapping upstroke and downstroke, respectively.

As illustrated in figure 18(a), the nominal lift coefficient ${C}_{L,n}$ of the moult-gapped wing with a moult gap near the root ( $d = -0.4$ ) matches that of the intact wing during $3.25 \lt {t/T} \lt 3.58$ and surpasses it for the rest of the period. This indicates that the aerodynamic benefit of this gap position arises primarily from downstroke deceleration ( $3.50\lt {t/T}\lt 3.75$ ) and upstroke. Table 4 shows that the lift increments of the moult-gapped wing over the intact wing during the upstroke deceleration ( $3.00\lt {t/T}\lt 3.25$ ), downstroke deceleration ( $3.50\lt {t/T}\lt 3.75$ ) and upstroke acceleration ( $3.75\lt {t/T}\lt 4.00$ ) stages are +0.09, +0.08 and +0.10, respectively. In contrast, the downstroke acceleration stage ( $3.25\lt {t/T}\lt 3.50$ ) exhibits a decrease of 0.03. The moult-gapped wing with the normalised gap position $d = 0.3$ outperforms the intact wing in terms of the average nominal lift coefficient, as shown in figure 17(b). This lift enhancement is attributed to the slower decay rate during the downstroke deceleration stage ( $3.50\lt {t/T}\lt 3.75$ ), resulting in a net positive increment in lift across the downstroke deceleration ( $3.50\lt {t/T}\lt 3.75$ ), upstroke acceleration ( $3.75\lt {t/T}\lt 4.00$ ) and upstroke deceleration ( $3.00\lt {t/T}\lt 3.25$ ) stages, compensating for the loss in the downstroke acceleration stage ( $3.25\lt {t/T}\lt 3.50$ ), as detailed in table 4. The nominal drag coefficients ${C}_{D,n}$ for moult-gapped wings with $d$ of −0.4 and 0.3 are approximately that of the intact wing during downstroke (see figure 18 b). The increased drag for these configurations primarily arises during upstroke, with average drag increments of +0.04 and +0.02, respectively, relative to the intact wing (see table 5).

Table 4.

Increments in the average nominal lift coefficient of the moult-gapped wing compared to the intact wing at different stages.

Table 5.

Increments in the average nominal drag coefficient of the moult-gapped wing compared to the intact wing at different stages.

As shown in figure 19(a), the nominal lift coefficient ${C}_{L,n}$ of the moult-gapped wing with a moult gap near the wing mid ( $d=0.1$ ) decreases below that of the intact wing during two intervals. Specifically, a major decrease occurs in the interval ${3.25\lt t}/{T}\lt 3.55$ , while a minor decrease appears in the interval ${3.70\lt t}/{T}\lt 3.80$ . The nominal lift coefficient ${C}_{L,n}$ for the moult-gapped wing with a wing-tip gap ( $d = 0.4$ ) is also less than that of the intact wing during the interval $3.25\lt {t}/{T}\lt 3.55$ , but exceeds it during the interval $3.60\lt {t}/{T}\lt 3.90$ . Thus, the lift loss results from a compromise between the decrease in lift during the downstroke acceleration ( $3.25\lt t/T\lt 3.50$ ) and the possible increase in lift near downstroke reversal. Table 4 shows that the increments in the average nominal lift coefficient for both the moult-gapped wings experience a greater decrease during the downstroke acceleration stage ( $3.25\lt t/T\lt 3.50$ ) than the total increase observed in the other three stages. Moreover, the average nominal drag coefficients $\overline {{C}_{D,n}}$ for the moult-gapped wings with normalised gap positions $d$ of 0.1 and 0.4 equal that of the intact wing (see figure 19 b), which results from balanced opposite increments near the maximum flapping velocities during downstroke (at the moment $t/T=3.50$ ) and upstroke (at the moment $t/T=4.00$ ).

Figure 19.

Instantaneous nominal aerodynamic force coefficients of the moult-gapped wings that lose the aerodynamic forces. (a) Nominal lift coefficient ${C}_{L,n}$ . (b) Nominal drag coefficient ${C}_{D,n}$ . The white and shaded blocks represent flapping upstroke and downstroke, respectively.

In summary, the lift of the moult-gapped wings depends on the trade-off between lift loss during downstroke acceleration ( $3.25\lt t/T\lt 3.50$ ) and lift enhancement in the subsequent three stages. Specifically, the decrease in lift near the lift peak and the increase or decrease in lift near downstroke reversal play crucial roles. Regarding the drag of the moult-gapped wings, the positive increment near the drag peak is dominant or, at the very least, balanced by the negative increment near the drag valley.

The spanwise position affects the relative velocity and interaction length between the fluid at the moult gap and the wing, regulating the generation and evolution of GVs. As the moult gap moves outwards along the span, there is an increase in the relative velocity between the fluid at the moult gap and the wing, coupled with a longer interaction length. Moult gaps located towards the outer wing generate GVs more frequently than those near the inner wing, resulting in more GVs shedding within a single stroke. This enhanced activity disrupts the attachment state of the LEVs, thereby reducing the vortex lift. The role of the local relative velocity in modulating the vortex forces through the GVs can also be revealed by the instantaneous nominal aerodynamic forces under different gap widths, as shown in figure 6. When the gap position is fixed, the changes in the local relative velocity stem from variations in the wing angular velocity caused by the elevation motion. In this scenario, the maximum local relative velocity occurs near the peak of the aerodynamic force, where the maximum lift loss is also produced.

The strengths and directions of the main vortex structures, such as the LEVs, are non-uniformly distributed along the span (Li & Nabawy Reference Li and Nabawy2024). For a typical rectangular wing, the flapping motion generates a bow-shaped LEV at the leading edge, indicating that the local direction of the LEVs changes along the span. The LEVs closely follow the straight leading edge of the wing at the wing mid. Towards the wing tip, they diverge from the leading edge owing to the influence of the TVs. At the wing root, the confined corner space forces the LEVs away from the leading edge. Overall, the vortex line representing the direction of the LEVs approximates an arcuate convex curve that bends towards the trailing edge, as depicted by the dotted line in figure 20(a). Moreover, it is equally important to recognise that the local strength of the LEVs varies along the span, peaking at the wing mid and dimnishing to its minimum value at the wing root and tip, as indicated by the size of the rotation symbol in figure 20(a). Consequently, the effect stemming from the interaction between the LEVs and GVs initially intensifies and then decreases as the moult gap moves from the wing root to the wing tip along the span. In this context, the effect of the interaction between LEVs and GVs on aerodynamic forces is most profoundly affected by the wing-mid moult gap. This is evidenced by the changes in LEVs during the downstroke acceleration stage ( $3.25\lt t/T\lt 3.50$ ), as illustrated in figure 20(b). The significant lift loss observed in the moult-gapped wing, with a normalised gap position of 0.1, is primarily attributed to this distinctive structural feature of LEVs.

Figure 20.

Spacial structures of the LEVs of the intact wing and the moult-gapped wing with a normalised gap position of 0.1. (a) Top view. (b) Back view. The vortex structures are represented by isosurfaces of $Q=8$ and coloured by the normalised spanwise vorticity ${\omega }_{y}$ . The flow structures in the figure appear at the moment ${t}/{T}=3.43$ . The spanwise variations of the strength and direction of the LEVs are presented schematically in panel (a). The area enclosed by the dashed line in panel (b) shows the strong interaction between the LEVs and GVs.

The local relative velocity at the moult gap accounts for the lift loss near the lift peak. The spanwise variation in the LEVs is associated with lift enhancement or loss near downstroke reversal. The change in flapping velocity direction near downstroke reversal intensifies the coupling between velocity and vorticity in the flow, which correlates with vortex lift through the Lamb vector. The Lamb vector is known as the origin of vortical flow nonlinearity and its normalised form is written as

(3.16) \begin{equation} \boldsymbol{l}=\boldsymbol{u} \times \boldsymbol{\omega }, \end{equation}

where $\boldsymbol{\omega }=\boldsymbol{\nabla }\times \boldsymbol{u}$ is the dimensionless vorticity vector normalised by ${2{{U}_{\infty }}}/{b}$ . The spatial integral of the vertical component of the Lamb vector, ${{l}_{z}}={{u}_{x}}{{\omega }_{y}}-{{u}_{y}}{{\omega }_{x}}$ , measures the magnitude of the lift in unsteady vortex dynamics, and its spatial distribution can be used to reveal the contribution of the main flow structures to lift generation (Marongiu, Tognaccini & Ueno Reference Marongiu, Tognaccini and Ueno2013; Wang et al. Reference Wang, Xu, Zhang and Wang2021, Reference Wang, Liu, Xu and Wang2022).

Figure 21.

Distribution of the vertical component of the normalised Lamb vector and its two decomposed parts. (a) Intact wing. (b) Moult-gapped wing (normalised gap position $d$ of −0.4). The flow structures in the figure appear at the downstroke reversal, ${t}/{T}=3.76$ . The spanwise distance of the slice across narrow sections of the moult-gapped wing from the symmetry plane at the wing root is 19 % of the semi-span $b/2$ .

Next, an analysis is conducted on the Lamb vector for the moult-gapped wing, specifically with the moult gap located close to the wing root ( $d=-0.4$ ), near downstroke reversal. Figure 21 shows the distributions of the vertical component of the Lamb vector $l_z$ of the moult-gapped wing, and its decomposition parts ${{l}_{z1}}={{u}_{x}}{{\omega }_{y}}$ and ${{l}_{z2}}=-{{u}_{y}}{{\omega }_{x}}$ at the moment ${t}/{T}=3.76$ . The inclination of the LEVs near the wing root produces spanwise and streamwise vorticities of the same order of magnitude. Both vorticity components contribute positively to the lift enhancement under the action of gap flow. For the intact wing in figure 21(a), the positive values of the vertical component $l_z$ of the Lamb vector in the near field are primarily concentrated in a strip region initiated at the leading edge that forms a specific angle with the upper wing surface. In contrast, the region with negative values is attached to the lower wing surface and rolls up downstream the trailing edge. Compared with the intact wing, the increment in the vertical component $l_z$ of the Lamb vector for the moult-gapped wing, with a moult gap proximate to the wing root, is predominantly situated in a thicker strip region. Additionally, there is an increased magnitude in the Lamb vector component $l_{z}$ in the region between the upper wing surface and strip region. These changes in and below the strip region stem from the decomposed parts $l_{z1}$ and $l_{z2}$ , respectively, as illustrated in figure 21(b). Further decomposition, as shown in figure 22, reveals that the thicker strip region results from the reduced recirculation zone and vertical diffusion of the spanwise vorticities of the LEVs because of gap leakage. Specifically, gap leakage not only augments the flow momentum of the separation zone above the upper wing surface, thereby reducing the recirculation velocity magnitude, but also intensifies the downwards diffusion of spanwise vorticities. Moreover, the gap leakage accumulates negative streamwise vorticities from the LEVs upstream of the leeward side, resulting in the increased magnitude of the vertical Lamb vector component in the region between the upper wing surface and strip region, as depicted in figure 22(b). The effect of the moult gap near the wing root on enhancing the lift by adjusting the LEVs is more pronounced during the upstroke acceleration stage ( $3.75\lt t/T\lt 4.00$ ). Similar Lamb vector analyses for moult gaps near the wing tip (normalised gap position of 0.3) and near the wing mid (normalised gap position of 0.1) are provided in Appendix B. The results are summarised in table 6. The circle mark ( $\bigcirc$ ) and cross mark ( $\times$ ) represent enabled and disabled actions, respectively.

Figure 22.

Distribution of the velocity and vorticity components at the downstroke reversal, ${t}/{T}=3.76$ . (a) Intact wing. (b) Moult-gapped wing (normalised gap position $d$ of −0.4). The spanwise distance of the slice across narrow sections of the moult-gapped wing from the symmetry plane at the wing root is 19 % of the semi-span $b/2$ .

Table 6.

Lift enhancement or loss of the moult-gapped wings with three different gap positions near downstroke reversal and identification of vorticity sources from the LEVs and TVs based on the Lamb vector analysis ( $\bigcirc$ , enabled; $\times$ , disabled).

When the moult gap is at the non-wing-tip position (normalised gap position $d\leqslant 0.3$ ), the flow topology around the moult-gapped wing differs from that of the original intact wing. By contrast, when the moult gap is at the wing tip (normalised gap position $d=0.4$ ), the moult-gapped wing reduces to the continuous shape of the intact wing, and the flow exhibits a topology analogous to that of the intact wing. The presence of a moult gap changes the shape of the local wing tip, which directly affects the formation and evolution of the TVs. Under normal conditions, many birds form gaps in their wing tips by adjusting the vertical positions of the front primary feathers. This wing-tip configuration, similar to a winglet (Whitcomb Reference Whitcomb1976), propagates vorticity in both the horizontal and vertical directions, which can significantly reduce induced drag during gliding flight (Tucker Reference Tucker1995). The effect of the winglet-like structure on the aerodynamic performance strongly depends on its specific configuration. The experimental example under a total gap size ratio of 0 given by Lynch, Mandadzhiev & Wissa (Reference Lynch, Mandadzhiev and Wissa2018) corresponds to the moult-gapped wing with $d=0.4$ in this study. Their results show that the lift coefficient of gliding flight could be significantly improved owing to the wing-tip gap. Nevertheless, in flapping flight cases, the lifting-surface loss is the dominant factor. As shown in figure 19(a), the wing with a wing-tip gap loses significant lift near the lift peak but increases the lift at downstroke reversal. Because of the larger lift loss during the former stage, the average nominal lift coefficient $\overline {{{C}_{L,n}}}$ decreases. In contrast, the average nominal drag coefficient $\overline {{{C}_{D,n}}}$ remains unchanged when compared with that of the intact wing.