3.1. Aerodynamic forces of the fixed wings with different local sweep angles

Figure 3 presents the distribution of the average aerodynamic force coefficients of fixed wings with different local sweep angles. According to the signs of the sweep angles of the inner and outer wings, the fixed wings with different local sweep angles are categorised into four types of configurations, including: (i) the configuration with backward-swept inner and outer wings $({ \Lambda }_{\textit{iw}}\gt 0, { \Lambda }_{\textit{ow}}\gt 0)$ ; (ii) the configuration with forward-swept inner wing and backward-swept outer wing $({ \Lambda }_{\textit{iw}}\lt 0, { \Lambda }_{\textit{ow}}\gt 0)$ ; (iii) the configuration with forward-swept inner and outer wings $({ \Lambda }_{\textit{iw}}\lt 0, { \Lambda }_{\textit{ow}}\lt 0)$ ; and (iv) the configuration with backward-swept inner wing and forward-swept outer wing $({ \Lambda }_{\textit{iw}}\gt 0, { \Lambda }_{\textit{ow}}\lt 0)$ , as illustrated in figure 3(a). These configurations are located in the four quadrant subspaces from I to IV, respectively. For each fixed-wing configuration, the aerodynamic force coefficients are averaged over one period of aerodynamic force oscillation when the flow reaches the fully developed unsteady separation state. The radial basis function is used to fit the distribution surface of the average aerodynamic force coefficients in figure 3(b–d). The dash-dotted line represents the configuration where the inner and outer wings have the same local sweep angle.

Figure 3.

Average aerodynamic force coefficients in the space $({ \Lambda }_{\textit{iw}}, { \Lambda }_{\textit{ow}})$ spanned by the local sweep angles of the inner and outer wings. (a) Schematic diagram of the four typical fixed wings with different local sweep angles. (b) Average lift coefficient. (c) Average drag coefficient. (d) Average lift-to-drag ratio. In panel (a), red and blue represent backward-swept (positive) and forward-swept (negative) local sweep angles, respectively. In panels (b–d), square and circular symbols denote the maximum average lift configuration, $({ \Lambda }_{\textit{iw}}, { \Lambda }_{\textit{ow}})=(10^{\circ }, 30^{\circ })$ , and its symmetric configuration, $({ \Lambda }_{\textit{iw}}, { \Lambda }_{\textit{ow}})=(-10^{\circ }, -30^{\circ })$ , respectively.

The configurations with the same local sweep angle change the aerodynamic force to a limited extent. As the same sweep angle increases (towards backward direction), the average lift and drag coefficients first increase and then decrease. In contrast, the different local sweep angles enable the fixed wings to change the aerodynamic force significantly, thereby extending their aerodynamic performance. The average lift coefficient varies from 0.63 to 0.83 and the average drag coefficient varies from 0.57 to 0.71, while the average lift-to-drag ratio varies from 1.01 to 1.20. Their ranges normalised by corresponding maximum values are 0.24, 0.20 and 0.16, respectively. The configuration with a backward-swept inner wing is more advantageous for generating high lift, as shown in figure 3(b). Moreover, the configuration with a backward-swept outer wing is more conducive to attaining a high lift-to-drag ratio, as exhibited in figure 3(d). For the investigated gliding fixed wings with different local sweep angles, the maximum average lift configuration, the minimum average drag configuration and the maximum average lift-to-drag ratio configuration are all in Quadrant I. The corresponding combinations of the local sweep angles of the inner and outer wings are $(10^{\circ }, 30^{\circ })$ , $(20^{\circ }, 20^{\circ })$ and $(10^{\circ }, 20^{\circ })$ , respectively. Mind that the point $(20^{\circ }, 20^{\circ })$ representing the minimum average drag configuration is on the right boundary of display range in figure 3(c).

In gliding flight, a high geometric angle of attack causes the flow separation over the fixed wing. The separated flow exhibits unsteadiness due to the generation of the LEVs and TEVs that evolve at a characteristic frequency (Taira & Colonius Reference Taira and Colonius2009; Kim & Gharib Reference Kim and Gharib2010; DeVoria & Mohseni Reference DeVoria and Mohseni2017). This flow unsteadiness differs from that in flapping flight, where the characteristic frequency of the separated flow couples with that of the flapping motion to produce complex unsteady aerodynamic effects (Trizila et al. Reference Trizila, Kang, Aono, Shyy and Visbal2011; Chin & Lentink Reference Chin and Lentink2016). The fixed wings with different local sweep angles exhibit varying degrees of unsteadiness, which can be reflected by the dimensionless principal amplitude and frequency of the time-dependent lift coefficient, as shown in figure 4. The dimensionless principal amplitude and frequency are obtained through the Fourier transform of the time-dependent lift coefficient sequence within the dimensionless time range of 200–300. To reduce frequency leakage, the Hanning window function is employed to handle the series of the time-dependent lift coefficient. The dimensionless principal frequency is derived from normalisation by the characteristic frequency ${{U}_{\infty }}/{{\gamma }{c}_{\textit{max}}}$ .

Figure 4.

Oscillation parameters of the time-dependent lift coefficient in the space $({ \Lambda }_{\textit{iw}}, { \Lambda }_{\textit{ow}})$ spanned by the local sweep angles of the inner and outer wings. (a) Dimensionless principal amplitude of the time-dependent lift coefficient. (b) Dimensionless principal frequency of the time-dependent lift coefficient. The square and circular symbols mark the maximum average lift configuration, $({ \Lambda }_{\textit{iw}}, { \Lambda }_{\textit{ow}})=(10^{\circ }, 30^{\circ })$ , and its symmetric configuration, $({ \Lambda }_{\textit{iw}}, { \Lambda }_{\textit{ow}})=(-10^{\circ }, -30^{\circ })$ , respectively.

The maximum average lift configuration exhibits the highest lift-coefficient amplitude of 0.11 and the highest lift-coefficient frequency of 0.38. The configurations with backward-swept inner wings (Quadrants I and IV) have higher amplitudes and frequencies of the lift coefficient compared with those with forward-swept inner wings (Quadrants II and III), indicating that the configurations with backward-swept inner wings enhance the unsteadiness. When the outer wing is also backward-swept, this unsteadiness is the most significant. The increased frequency of the backward-swept configurations relative to the unswept configurations is consistent with the experimental measurements of Huang & Lin (Reference Huang and Lin1995) and Yen & Hsu (Reference Yen and Hsu2007). On one hand, to enhance the average lift coefficient, configurations with backward-swept inner and outer wings are recommended. On the other hand, to reduce the lift coefficient oscillations, these same configurations should be avoided. This inherent trade-off means that the high average lift and low oscillating lift represent a contradiction in the gliding flight.

3.2. Unsteady lift of the local sweep morphing

We focus on the maximum average lift configuration, $({ \Lambda }_{\textit{iw}}, { \Lambda }_{\textit{ow}})=(10^{\circ }, 30^{\circ })$ , and its symmetric configuration with the local sweep angles of equal values, but opposite signs, $({ \Lambda }_{\textit{iw}}, { \Lambda }_{\textit{ow}})=(-10^{\circ }, -30^{\circ })$ . These two configurations are denoted by square and circular symbols, respectively, in figures 3 and 4. The average lift coefficients of the maximum average lift configuration and its symmetric configuration are 0.83 and 0.69, respectively. The time-dependent lift coefficient of the maximum average lift configuration has a dimensionless principal amplitude of 0.11 and a dimensionless principal frequency of 0.38. In contrast, that of the symmetric configuration has a dimensionless principal amplitude of 0.01 and a dimensionless principal frequency of 0.28.

The maximum average lift configuration generates a high average lift, while the symmetric configuration generates a low oscillating lift, representing two typical aerodynamic requirements. The local sweep morphing between the two configurations provides a solution to compromise the high average lift and low oscillating lift. The local sweep morphing in both the forward and backward directions can be achieved through actuations inspired by the shoulder and wrist of the avian wing, as shown in figure 5. Hereinafter, the pre-morphing wings of the forward and backward sweeping are consistently taken as the maximum average lift configuration and its symmetric counterpart, respectively.

Figure 5.

Local sweep morphing between the maximum average lift configuration, $({ \Lambda }_{\textit{iw}}, { \Lambda }_{\textit{ow}})=(10^{\circ }, 30^{\circ })$ , and its symmetric configuration, $({ \Lambda }_{\textit{iw}}, { \Lambda }_{\textit{ow}})=(-10^{\circ }, -30^{\circ })$ : (a) forward sweeping; (b) backward sweeping. The solid and dashed outlines denote the configurations after and before morphing, respectively.

The controlled evolution of lift during the local sweep morphing is challenging, because the local sweep morphing initiates and terminates with distinct unsteady separated flows. Figure 6 illustrates that the fully developed lift coefficients undergo periodic oscillations with a single harmonic component, but with different amplitudes and frequencies. The oscillating lift coefficients, each with a principle frequency, correspond to the periodic unsteady type of Taira & Colonius (Reference Taira and Colonius2009), according to the given angle of attack of and aspect ratio. The Strouhal number, based on the principle frequency, projected chord length ( ${{c}_{\textit{max}}}\sin \alpha$ ) and free stream velocity, is 0.13 and 0.10 for the maximum average lift configuration and its symmetric configuration, respectively. These two values approach the converged Strouhal number of 0.12 from all periodic shedding cases under similar conditions. The maximum average lift configuration has a high dimensionless principal amplitude and a high dimensionless principal frequency, while the situation is the opposite for its symmetric configuration. The weak unsteadiness of the symmetric configuration may be related to the downwash effect of the tip vortices (Zhang & Taira Reference Zhang and Taira2022). The local sweep morphing requires to switch between two flows with such different unsteadiness.

Figure 6.

Lift-coefficient history of two typical configurations to experience the local sweep morphing.

Figure 7( a) displays the time-dependent lift coefficients during the forward sweeping under three morphing kinematics, with the morphing start of the valley moment and the morphing duration of twice the pre-morphing period ( $T=2.60$ ). The three morphing kinematics are in the step, linear and nonlinear sine forms, as shown in figure 2. The morphing phase $[{{t}_{0}},{{t}_{f}}]$ is magnified and highlighted by a shaded block. The lift characteristics of forward sweeping under the three morphing kinematics are summarised in Appendix C. For the step morphing kinematics, the lift coefficient experiences a steep change immediately after the morphing. Under the linear morphing kinematics, the lift coefficient also undergoes steep changes at the beginning and end of the morphing. These changes are notably reduced compared with those of step morphing. In contrast, the nonlinear sine morphing completely eliminates the steep change in the lift coefficient due to its smooth nature. However, the lift coefficient increases substantially during the morphing phase, leading to a lift overshoot. The local maximum lift coefficient increases by 60 % compared with the average lift coefficient of 0.83 before the morphing. The unsteady characteristics after the forward sweeping are the recurrence and decay of the lift peaks. This behaviour is similar to the lift coefficient response of the streamwise acceleration of the flat-plate wing under the same geometric angle of attack (Mulleners et al. Reference Mulleners, Mancini and Jones2017). It takes more than 10 convective time scales to reach the new unsteady state.

Figure 7.

Effect of the morphing kinematics on the time-dependent lift coefficient: (a) forward sweeping; (b) backward sweeping.

The effect of morphing kinematics on the time-dependent lift coefficient during the backward morphing is shown in figure 7( b). The same three morphing kinematics are applied with the morphing start of the valley moment and the morphing duration of twice the pre-morphing period ( $T=3.52$ ). The corresponding lift characteristics are presented in Appendix C. Compared with the forward sweeping, the lift coefficients of the step and linear morphing kinematics in the backward sweeping exhibit less steep changes. The lift coefficient of the nonlinear sine morphing decreases significantly during the morphing, resulting in a lift undershoot. The local minimum lift coefficient decreases by 27 % compared with the pre-morphing average lift coefficient of 0.69. The lift coefficient after the backward sweeping is characterised by the recurrence and growth of the lift peaks. Similarly, more than 10 convective time scales are required to arrive at the new unsteady state. The lift undershoot during the backward sweeping may cause the stall-like issues. The maximum lift undershoot has a smaller magnitude compared with the maximum lift overshoot. This difference arises from the combined effects of the sweeping direction and pre-morphing unsteadiness.

The lift over/undershoot is expected to be alleviated by adjusting the morphing start and duration. Figure 8 illustrates the effect of the morphing start on the time-dependent lift coefficient. The nonlinear sine morphing kinematics is adopted and the morphing duration is fixed at twice the pre-morphing period. Four typical morphing starts including the peak ( ${t}_{p}$ ), descending-mean ( ${t}_{d}$ ), valley ( ${t}_{v}$ ) and ascending-mean ( ${t}_{a}$ ) moments are compared. We take the morphing start of the valley moment as the reference. To align the morphing phase, we use the scaled time relative to the morphing start, $(t-t_{0})/T$ , here referred to as scaled relative time. The comparison in Appendix C shows that starting at the valley moment can achieve the most pronounced reduction in the maximum lift overshoot during the forward sweeping. In contrast, adjusting the morphing start cannot effectively reduce the lift undershoot during the backward sweeping, due to the weak unsteadiness present before this sweeping.

Figure 8.

Effect of the morphing start on the time-dependent lift coefficient: (a) forward sweeping; (b) backward sweeping. The nonlinear sine morphing kinematics is employed with the morphing duration of $2T$ and the morphing starts of the peak ( $t_p$ ), descending-mean ( $t_d$ ), valley ( $t_v$ ) and ascending-mean ( $t_a$ ) moments. The morphing phase is aligned using the scaled relative time.

Figure 9 presents the time-dependent lift coefficients under different morphing durations. With the same nonlinear sine morphing kinematics, the forward and backward sweeping wings start morphing from the valley and peak moments, respectively. Four typical morphing durations spanning from one to four times the pre-morphing lift coefficient period $T$ are compared. We use the morphing duration of $2T$ as the reference. For the forward sweeping in figure 9( a), when the morphing duration is shortened from $2T$ to $T$ , the maximum lift coefficient arises significantly, giving rise to a severe lift overshoot. When the morphing duration is prolonged from $2T$ to $3T$ or $4T$ , the maximum lift coefficient experiences a decrease. The lift characteristics during the forward sweeping are also observed in the accelerating start of the flat-plate wing (Mancini et al. Reference Mancini, Manar, Granlund, Ol and Jones2015; Stevens et al. Reference Stevens2017). Similarly, during the backward sweeping in figure 9( b), shortening the morphing duration increases the lift undershoot, while prolonging the morphing duration decreases the lift undershoot.

Figure 9.

Effect of the morphing duration on the time-dependent lift coefficient: (a) forward sweeping; (b) backward sweeping. The nonlinear sine morphing kinematics is employed with the morphing start of the valley ( $t_v$ ) and peak ( $t_p$ ) moments for the forward and backward sweeping cases, respectively. The morphing durations ranging from $T$ to $4T$ are adopted.

Figure 10 plots the lift-coefficient hysteresis loop of a favourable local sweep morphing. The favourable morphing strategy is achieved using the nonlinear sine kinematics with the morphing start of the valley moment and the morphing duration of $3T$ during the forward sweeping, and the same kinematics with the morphing start of the peak moment during the backward sweeping. The lift-coefficient branches of the forward sweeping and the backward sweeping are denoted by the red solid and blue dashed lines, respectively. The morphing process is indicated by the local sweep angle ${ \Lambda }_{\textit{iw}}$ of the inner wing. To clarify the difference of the lift coefficient between the sweeping wing (dynamic) and fixed-wing (static) cases, the oscillating ranges of three fixed-wings are represented by the rectangle columns. The left and right columns denote the pre-morphing fixed-wings of backward sweeping ( ${{ \Lambda }_{\textit{iw}}}=-10{}^\circ$ , ${{ \Lambda }_{\textit{ow}}}=-30{}^\circ$ ) and forward sweeping ( ${ \Lambda }_{\textit{iw}}=10{}^\circ$ , ${{ \Lambda }_{\textit{ow}}}=30{}^\circ$ ), respectively. The middle column corresponds to the unswept fixed-wing, a configuration traversed during the morphing process.

Figure 10.

Hysteresis loop of the lift coefficient under the favourable local sweep morphing. Red solid line (forward sweeping): the nonlinear sine kinematics with the morphing start of the valley moment $t_v$ and the morphing duration of $3T$ . Blue dashed line (backward sweeping): the nonlinear sine morphing kinematics with the morphing start of the peak moment $t_p$ and the morphing duration of $3T$ . The top and bottom of the rectangle column display the maximum and minimum lift coefficients, respectively, of the fixed wing with variable local sweep angles. From left to right, the local sweep angles $({ \Lambda }_{\textit{iw}}, { \Lambda }_{\textit{ow}})$ are $(-10^{\circ }, -30^{\circ })$ , $(0^{\circ }, 0^{\circ })$ and $(10^{\circ }, 30^{\circ })$ .

As shown in figure 10, the lift coefficient of the forward sweeping is higher than that of the backward sweeping during the local sweep morphing. The lift oscillation of the forward sweeping is stronger than that of the backward sweeping, which is a continuation of the highly unsteady initial flow. After the forward sweeping, a relaxation time of lift decay (denoted by the downward red arrow) is required to return to the fully developed state. Conversely, a growth relaxation (denoted by the upward blue arrow) occurs after the backward sweeping. These characteristics are distinct from those of the sweep morphing at the low geometric angles of attack, which aligns with the steady initial flow (Zeng et al. Reference Zeng, Liu, Shao and Li2023; Bai et al. Reference Bai, Yang, Guo, Xiao, Liu and Deng2025). Furthermore, the mentioned differences in the time-dependent lifts of the forward and backward sweeping cases are caused by the hysteresis nature of the flow. The lift hysteresis of the morphing wing is owing to the fact that the flow cannot immediately respond to the changes in the morphing motion of the wing, which is similar to the pitching situation (Mittal & Saxena Reference Mittal and Saxena2002).

3.3. Sweeping modification of the lift over/undershoot

The time-dependent lift coefficient during the local sweep morphing, especially the lift over/undershoot, depends on the pre-morphing flow state. Furthermore, the lift coefficient profiles under the forward-sweeping and no-morphing cases exhibit similar shapes, both featuring two peaks, although their magnitudes are different, as displayed in figure 11( a). This similarity implies that one component of the lift coefficient during the morphing is extended from the pre-morphing lift coefficient. When the extended lift coefficient is subtracted from the total lift coefficient during the morphing, the net lift coefficient attributes to the modification from the local sweep morphing, as shown in figure 11( b). We refer to the net component as the sweeping-modification lift coefficient.

Figure 11.

Decomposition of the time-dependent lift coefficient during the forward sweeping: (a) lift coefficients under the forward sweeping and no sweeping; (b) sweeping-modification lift coefficient. The forward-sweeping morphing phase is visualised by the shaded block. The negligible shift from zero at the morphing start results from the filtering of the lift coefficient.

Figure 12 presents the dependence of the sweeping-modification lift coefficient on the morphing start. The sweeping-modification lift coefficient is plotted against the scaled relative time. For the forward sweeping, the sweeping-modification lift coefficients exhibit similar convex profiles under different morphing starts, as shown in figure 12( a). In the first 30 % of the morphing phase, the sweeping-modification lift coefficients almost overlap, indicating an interval in which the sweeping-modification lift coefficient is independent of the morphing start. Mancini et al. (Reference Mancini, Manar, Granlund, Ol and Jones2015) also report the similarity in the measured lift coefficients during the accelerating start of a flat-plate wing. They find that the circulatory lift component is not affected by the acceleration profile in the first 40 % of the accelerating phase. In the remaining part of the forward sweeping, deviations occur in the sweeping-modification lift coefficients. For the backward sweeping, the sweeping-modification lift coefficients under different morphing starts have a consistent concave profile and overlap throughout the morphing phase, as displayed in figure 12( b). The similarity of the sweeping-modification lift coefficients facilitates the modelling of the time-dependent lift over/undershoot from the local sweep morphing.

Figure 12.

Effect of the morphing start on the sweeping-modification lift coefficient: (a) forward sweeping; (b) backward sweeping. The vertical dashed line represents the mid-moment of the morphing phase.

Figure 13 illustrates the effect of the morphing duration on the sweep-modification lift coefficient during the forward sweeping. Two sweeping modes are distinguished by the morphing duration of $2T$ . A fast sweeping mode corresponds to the morphing duration ${{t}_{f}}-{{t}_{0}}\in [T, 2T]$ , while a slow sweeping mode corresponds to the morphing duration ${{t}_{f}}-{{t}_{0}}\in (2T, 4T]$ . For the fast sweeping mode in figure 13( a), the sweeping-modification lift coefficient almost reaches its peak value at mid-moment of the morphing phase. Before and after this peak value, the sweeping-modification lift coefficient shows a monotonic ascent and a monotonic descent, respectively. As the morphing duration increases from $T$ to $2T$ , the growth of the lift coefficient in the ascending stage and its decay in the descending stage both slow down, but with different rates. The profile of the sweeping-modification lift coefficient changes from a near-symmetric shape to an asymmetric one. As will be explained later, the asymmetric profile consists of a symmetric basic component and an asymmetric variation, which correlate with the horizontal and vertical components of the morphing velocity, respectively.

Figure 13.

Effect of the morphing duration on the sweeping-modification lift coefficient during the forward sweeping: (a) fast sweeping mode with the morphing duration ${{t}_{f}}-{{t}_{0}}\in [T, 2T]$ ; (b) slow sweeping mode with the morphing duration ${{t}_{f}}-{{t}_{0}}\in (2T, 4T]$ .

By contrast, the slow sweeping mode is characterised by a new extreme value on the descending branch of the sweeping-modification lift coefficient (see figure 13 b). As the morphing duration increases, this lift extreme develops into a secondary lift peak. When the morphing duration reaches $4T$ , several successive lift peaks appear in the descending stage due to the relaxation from the slow deceleration. From a decomposition perspective, the fast sweeping mode reflects a mechanism that constitutes the framework of the sweeping-modification lift coefficient, while the slow sweeping mode adjusts its descending branch by introducing additional unsteady effects.

Figure 14 presents the change in the maximum magnitude of the sweeping-modification lift coefficient with the morphing duration. There is a nonlinear decrease in the maximum sweeping-modification lift coefficient ${C}_{L, \Lambda , max }$ during the forward sweeping, as shown in figure 14( a). The maximum sweeping-modification lift coefficient decreases fast under the fast sweeping mode with the morphing duration ${{t}_{f}}-{{t}_{0}}\in [T, 2T]$ , while it decreases slowly under the slow sweeping mode with the morphing duration ${{t}_{f}}-{{t}_{0}}\in (2T, 4T]$ . Similarly, the minimum sweeping-modification lift coefficient ${C}_{L, \Lambda , min }$ during the backward sweeping exhibits a nonlinear increase with different increase rates under the fast and slow sweeping modes. The nonlinear changes follow the negative power laws,

(3.1) \begin{align} {{C}_{L, \Lambda , max }} & ={{C}_{ \Lambda -}}{{\left ( \frac {t_f-{{t}_{0}}}{T} \right )}^{-1}}, \\[-12pt]\nonumber \end{align}

(3.2) \begin{align} {{C}_{L, \Lambda , min }} & ={{C}_{ \Lambda +}}{{\left ( \frac {t_f-{{t}_{0}}}{T} \right )}^{-1}}, \end{align}

Figure 14.

Effect of the morphing duration on the maximum magnitude of the sweeping-modification lift coefficient. (a) Maximum sweeping-modification lift coefficient during the forward sweeping. (b) Minimum sweeping-modification lift coefficient during the backward sweeping. The circular symbols denote the results of numerical simulations, while the solid lines represent the fitted negative power laws.

where ${C}_{ \Lambda -}=1.0$ and ${C}_{ \Lambda +}=-0.4$ , measuring the change rates of the maximum and minimum sweeping-modification lift coefficients, respectively.

Through the maximum and minimum sweeping-modification lift coefficients, the sweeping-modification lift coefficients under the forward and backward sweeping cases are both scaled to the interval of $[0,1]$ . As shown in figure 15, for the forward sweeping, the scaled sweeping-modification lift coefficients, $C_{L, \Lambda }^{*}={{{C}_{L, \Lambda }}}/{{{C}_{L, \Lambda , max }}}$ , exhibit similarity among different morphing durations, especially within the ascending stage, ${ ( t-{{t}_{0}} )}/{ ( {{t}_{f}}-{{t}_{0}} )}\in [ 0,0.5 ]$ . The lift coefficient during the descending stage shows different divergences between the fast and slow sweeping modes. Higher similarity of the scaled sweeping-modification lift coefficients is found during the backward sweeping, as displayed in Appendix D. The similarity of the scaled sweeping-modification lift coefficients under different morphing durations paves the way for the modelling of time-dependent lift over/undershoot from local sweep morphing.

Figure 15.

Scaled sweeping-modification lift coefficients under different forward sweeping durations: (a) fast sweeping mode; (b) slow sweeping mode.

3.4. Reduced-order unsteady lift model for the local sweep morphing

3.4.1. Kinematic model of the local sweep morphing wing

As the foundation for modelling the lift over/undershoot, it is essential to propose a kinematic model of the local sweep morphing wing. We define the straight-line segment connecting the two points of the wing root and the wing tip on the half-chord line as the effective half-chord line, as shown in figure 16. For the wing with the inner-wing sweep angle ${ \Lambda }_{\textit{iw}}$ and the outer-wing sweep angle ${ \Lambda }_{\textit{ow}}$ , the effective sweep angle ${ \Lambda }_{e}$ is defined here as the angle between the effective half-chord line and the spanwise axis (Anderson Reference Anderson2010), and is expressed as

(3.3) \begin{equation} {{ \Lambda }_{e}}={{\tan }^{-1}}\left ( \frac {\tan {{ \Lambda }_{\textit{i}w}}+\tan {{ \Lambda }_{\textit{ow}}}}{2} \right ). \end{equation}

The local sweep angles $({{ \Lambda }_{\textit{iw}}},{{ \Lambda }_{\textit{ow}}})$ of the maximum average lift configuration are $(10^{\circ },30^{\circ })$ , leading to its effective sweep angle of ${21}^{\circ }$ . The corresponding symmetric configuration has the effective sweep angle with equal magnitudes and opposite signs.

Figure 16.

Schematic diagram of the kinematical model of the local sweep morphing wing: (a) three-dimensional view; (b) in-plane view; (c) left side view.

The nonlinear sine morphing kinematics in terms of the effective sweep angle is

(3.4) \begin{equation} {{ \Lambda }_{e}}(t)=\frac {{{ \Lambda }_{e,f}}-{{ \Lambda }_{e,0}}}{2}\sin \left [ \frac {\pi }{{{t}_{f}}-{{t}_{0}}}t-\frac {\pi \left ( {{t}_{f}}+{{t}_{0}} \right )}{2\left ( {{t}_{f}}-{{t}_{0}} \right )} \right ]+\frac {{{ \Lambda }_{e,f}}+{{ \Lambda }_{e,0}}}{2}, \end{equation}

where ${ \Lambda }_{e,0}$ and ${ \Lambda }_{e,f}$ are the initial and final effective sweep angles, respectively. For the forward sweeping, ${{ \Lambda }_{e,0}}={21}^{\circ }$ and ${{ \Lambda }_{e,f}}=-{21}^{\circ }$ .

The time derivative of the effective sweep angle is

(3.5) \begin{equation} \frac {\textrm{d}{{ \Lambda }_{e}}(t)}{\textrm{d}t}=\frac {{{ \Lambda }_{e,f}}-{{ \Lambda }_{e,0}}}{2}\cos \left [ \frac {\pi }{{{t}_{f}}-{{t}_{0}}}t-\frac {\pi \left ( {{t}_{f}}+{{t}_{0}} \right )}{2\left ( {{t}_{f}}-{{t}_{0}} \right )} \right ] \frac {\pi }{{{t}_{f}}-{{t}_{0}}}. \end{equation}

With the effective sweep angle, the sweep morphing in the shearing way results in the effective tangential displacement $\Delta {{s}_{e}}$ at a spanwise position $y$ ( $0 \leqslant {y} \leqslant {b}_{\textit{iw}}+{b}_{\textit{ow}}$ ) as follows:

(3.6) \begin{equation} \Delta {s_e} (t) = y\left [ \tan {{ \Lambda }_{e}}\left ( t \right )-\tan {{ \Lambda }_{e}}\left ( {{t}_{0}} \right ) \right ], \quad {{t}_{0}}\leqslant t\leqslant {{t}_{f}}. \end{equation}

According to the definition of the effective tangential displacement, the sweep morphing velocity ${U}_{s}$ at the spanwise position $y$ is

(3.7) \begin{equation} {U}_{s}\left ( t \right )={\frac {{U}_{\infty }y}{{\gamma }{c}_{\textit{max}}}}\frac {1}{{{\cos }^{2}}\left [ {{ \Lambda }_{e}}\left ( t \right ) \right ]}\frac {\textrm{d}{{ \Lambda }_{e}}\left ( t \right )}{\textrm{d}t}. \end{equation}

The net forward-flight velocity of the wing section at the spanwise position $y$ is

(3.8) \begin{equation} {{U}_{x}}\left ( t \right )={{U}_{\infty }}-{{U}_{s}}\left ( t \right )\cos \alpha. \end{equation}

The normal component of the net forward-flight velocity (see figure 17) perpendicular to the effective half-chord line is

(3.9) \begin{equation} {{U}_{n}}\left ( t \right )={{U}_{x}}\left ( t \right )\cos {{ \Lambda }_{e}}\left ( t \right ). \end{equation}

Figure 17.

Normal component of the net forward-flight velocity changes with the morphing duration: (a) forward sweeping; (b) backward sweeping. The vertical axis range of panel (a) is twice that of panel (b).

Here, the spanwise position $y$ is selected at the wing tip, namely, $y={b}_{\textit{iw}}+{b}_{\textit{ow}}$ . This is because the wing tip typically serves as the characteristic spanwise position to measure the unsteady effects of bio-locomotion (Taylor, Nudds & Thomas Reference Taylor, Nudds and Thomas2003).

3.4.2. Lift model of the local sweep morphing wing

We start to develop a reduced-order lift model for local sweep morphing wings, which facilitates low-cost lift evaluation during their conceptual design and feedback control (Brunton et al. Reference Brunton, Rowley and Williams2013). Previously, Manar & Jones (Reference Manar and Jones2019) extend the classical thin aerofoil theory to the cases of high geometric angles of attack and variable translational velocities, and obtain a reduced-order unsteady model for predicting the lift coefficient of a linearly accelerating flat-plate wing. However, this model is shown to overestimate the local overshoot of the normal force on the flat plate. To take into account the effects of a finite wingspan, we adopt the same assumptions to extend Prandtl’s lifting-line theory and propose a time-dependent lift model for a three-dimensional wing with the variable translational velocity. The detailed derivation of this model is presented in Appendix E.

The lift-curve slope of the proposed model is

(3.10) \begin{equation} {{\kappa }_{3\text{-}\textrm{D}}}=\frac {{{\kappa }_{{2\text{-}\textrm{D}}}}}{E+\displaystyle \frac {{{\kappa }_{{2\text{-}\textrm{D}}}}}{\pi {AR}}{{\left ( \frac {{{U}_{\textit{ref}}}}{{{U}_{w\textit{ing}}}} \right )}^{2}}\left ( 1+\delta \right )}, \end{equation}

where ${\kappa }_{{2\text{-}\textrm{D}}}$ is the lift-curve slope of a two-dimensional flat-plate aerofoil, $E$ is the edge velocity correction factor, $ AR$ is the aspect ratio of the unilateral wing (see Appendix A), ${U}_{w\textit{ing}}$ is the translational velocity of the forward-flight wing, and ${U}_{\textit{ref}}$ is the reference velocity for defining the lift coefficient in (2.8) and (2.9). Here, $\delta$ is the Glauert correction factor for the induced angle of attack, which accounts for the effects of a non-elliptical circulation distribution. The proposed model (3.10) indicates that the translational velocity affects the lift coefficient in the form of kinetic energy.

In this work, the two-dimensional lift-curve slope ${\kappa }_{{2\text{-}\textrm{D}}}$ is given as 5.16 to account for the Reynolds number effect (Okamoto, Yasuda & Azuma Reference Okamoto, Yasuda and Azuma1996; Nabawy & Crowther Reference Nabawy and Crowther2014). The edge velocity correction factor and Glauert correction factor are formulated to incorporate the effect of the sweep angle. With reference to the treatment of Polhamus (Reference Polhamus1949), the edge velocity correction factor ${E}_{ \Lambda }$ of the wing with a sweep angle $ \Lambda$ can be obtained by comparing the correction equations of Jones (Reference Jones1941) and Küchemann (Reference Küchemann1952), that is,

(3.11) \begin{equation} {{E}_{ \Lambda }}=\sqrt {\frac {1}{{{\cos }^{2}} \Lambda }+{{\left ( \frac {{{\kappa }_{{2\text{-}\textrm{D}}}}}{\pi {{AR}}} \right )}^{2}}}. \end{equation}

The Glauert correction factor ${\delta }_{ \Lambda }$ for the wing with a sweep angle $ \Lambda$ is determined according to the theoretical data of DeYoung (Reference DeYoung1947). Due to the differences in the Reynolds number and aspect ratio, the data of lift-curve slope are corrected using the relation by Gutierrez-Castillo et al. (Reference Gutierrez-Castillo, Aguilar-Cabello, Alcalde-Morales, Parras and del Pino2021), expressed as

(3.12) \begin{equation} {{\kappa}_{3\text{-}\textrm{D}}} \propto {{\left ( 1+\frac {{{10}^{6}}}{{Re}} \right )}^{-\sigma }}\left [ \sqrt {1+{{\left ( \frac {\lambda }{\pi {{AR}}} \right )}^{2}}}+\frac {\lambda }{\pi {{AR}}} \right ]^{-1}, \end{equation}

where the parameter $\lambda$ is set to its original value of 3.79. The increase rate $\sigma$ is decreased slightly to $1/6$ , since the sensitivity of the lift coefficient reduces at high Reynolds number (Quinn Jr Reference Quinn1945). The corrected lift-curve slope corresponds to the Reynolds number of $O(10^4)$ , which is the minimum order of magnitude allowed by the relation (3.12). The Glauert correction factor is calculated by substituting the corrected lift-curve slope into (3.10) that involves (3.11) and excludes the translational velocity of the wing. The dependence of the Glauert correction factor on the sweep angle $ \Lambda$ is approximated by

(3.13) \begin{equation} {\delta }_{ \Lambda } = {C_{\delta 1}} \left ( \frac {\pi }{180}{ \Lambda }-{C_{\delta 2}} \right )^2 + {C_{\delta 3}}, \end{equation}

where the constant parameters are taken as ${C_{\delta 1}}=0.13$ , ${C_{\delta 2}}=0.08$ and ${C_{\delta 3}}=0.17$ . This fitted formula coincides with the data for the sweep angles ranging from $-30^\circ$ to $30^\circ$ .

For the local sweep morphing wing, the sweep angle $ \Lambda$ in (3.11) and (3.13) is the effective sweep angle ${ \Lambda }_{e}$ , and the translational velocity ${U}_{w\textit{ing}}$ of the wing is the normal component ${U}_{n}$ of the net forward-flight velocity perpendicular to the effective half-chord line, as expressed in (3.9). Substituting (3.9), (3.11) and (3.13) into the model (3.10) yields the lift-curve slope formula applicable to the swept wing,

(3.14) \begin{equation} {{\kappa }_{{3\text{-}\textrm{D}}, \Lambda }}=\frac {{{\kappa }_{{2\text{-}\textrm{D}}}}}{\displaystyle {{E}_{ \Lambda }}+\frac {{{\kappa }_{{2\text{-}\textrm{D}}}}}{{\pi} {{AR}}}{{\left ( \frac {{{U}_{\infty }}}{{{U}_{n}}} \right )}^{2}} \left ( 1+{\delta }_{ \Lambda } \right )}. \end{equation}

The lift coefficient at high geometric angles of attack can be derived from that at low geometric angles of attack for the unsteady flows dominated by the LEVs (Nabawy & Crowther Reference Nabawy and Crowther2014; Wang et al. Reference Wang, Goosen and van Keulen2016), as detailed in Appendix F. We follow this approach of constructing the lift model at high geometric angles of attack and obtain the lift coefficient of the swept wing with variable translational velocity,

(3.15) \begin{equation} {{{\hat C}^{\prime}}_{L, \Lambda }} = \frac {1}{2} \frac {{{\kappa }_{{2\text{-}\textrm{D}}}}}{\displaystyle {E}_{ \Lambda }+\frac {{{\kappa }_{{2\text{-}\textrm{D}}}}}{{\pi} {{AR}}}{{\left ( \frac {{{U}_{\infty }}}{{{U}_{n}}} \right )}^{2}} \left ( 1+{\delta }_{ \Lambda } \right )} \sin \left ( 2\alpha \right ). \end{equation}

Here and hereinafter, the interim lift coefficient evaluated by the different lifting-line models is marked with a hat symbol to distinguish it from the final one.

To capture the key unsteady effects of the vortical flow, an unsteady Kutta–Joukowski theorem (Gennaretti & Giansante Reference Gennaretti and Giansante2022; Giansante, Bernardini & Gennaretti Reference Giansante, Bernardini and Gennaretti2024; Frassoldati et al. Reference Frassoldati, Giansante, Bernardini and Gennaretti2024) is incorporated into the current theoretical framework, as presented in Appendix G. Through the transformation and simplification of a frequency response function correlating the sectional lift and circulation, the unsteady Kutta–Joukowski theorem can be explicitly represented in the unsteady formulation of the model (3.15),

(3.16) \begin{equation} {{{\hat C}^{\prime \prime}}_{L, \Lambda }}( {t} )=\frac {1}{2}\frac {{{\kappa }_{{2\text{-}\textrm{D}}}}}{\displaystyle {{E}_{ \Lambda }}\left ( {t} \right )+\frac {{{\kappa }_{{2\text{-}\textrm{D}}}}}{{\pi} {{AR}}}{{\left [ \frac {{{U}_{\infty }}}{{{U}_{n}}\left ( {t} \right )} \right ]}^{2}} \frac {1+{\delta }_{ \Lambda }(t)}{{\varPhi }(t)}}\sin \left ( 2\alpha \right ), \end{equation}

where $\varPhi$ is an unsteady indicator function, which quantifies the wake memory effect associated with the unsteady evolution and interaction of vortex structures.

In this work, (3.16) is intended for two purposes. One is to construct a model for predicting the lift over/undershoot induced by the local sweep morphing and the other is to quantify the unsteady effects of local sweep morphing. For the unsteady flow dominated by a single principal frequency, the unsteady indicator function takes the specific form of

(3.17) \begin{equation} {\varPhi }_{k_T}(t) \simeq {C}_{k_T} =\sqrt {1+k^2_T}, \end{equation}

where $k_T$ is a reduced frequency, expressed as

(3.18) \begin{equation} k_T=\frac {\pi {{c}_{\textit{max}}^{*}}}{T{{U}_{\infty }^{*}}}, \end{equation}

where the quantities with an asterisk as the superscript are non-dimensionalised. The principal frequency ( $1/T$ ) of the pre-morphing lift coefficient is adopted, since the lift over/undershoot during the local sweep morphing is closely related to the pre-morphing unsteadiness of the flow, as shown in § 3.2.

It is noticed that the unsteady Kutta–Joukowski theorem (G8) reduces to the steady theorem multiplied by a constant scale factor ${C}_{k_T}$ . We would like to point out that it is appropriate to close the lift model through the steady theorem, when the unsteady effects induced by the local sweep morphing are sufficiently weak. In this sense, the reduced form ${C}_{k_T}$ of the unsteady indicator function measures the relevant unsteady effects based on the principal frequency. It corrects the steady Kutta–Joukowski theorem to the unsteady counterpart by taking the time derivative of the sectional circulation into account, as explained in Appendix G. The steady Kutta–Joukowski theorem is recovered when the unsteady indicator function ${C}_{k_T}=1$ . Numerical values for calculating the unsteady indicator function ${C}_{k_T}$ are shown in table 3. The unsteady indicator function ${C}_{k_T}$ equals 1.31 for the forward sweeping and 1.18 for the backward sweeping. The increases of 31 % and 18 % in the unsteady indicator function for the forward and backward sweeping cases, respectively, indicate that the unsteady theorem reflects non-negligible unsteady effects as compared with the steady theorem. Additionally, the unsteady indicator function ${C}_{k_T}$ in terms of the reduced frequency $k_T$ captures the discrepancies in lift generation associated with the pre-morphing flow states between the forward and backward sweeping cases. We would like to note that the reduced form ${C}_{k_T}$ is not permitted for flows characterised by multiple-frequencies, being applicable exclusively to the single-frequency scenario. Although a certain phase difference between sectional lift and circulation is induced at the values of the reduced frequency (Gennaretti & Giansante Reference Gennaretti and Giansante2022), we consider only the magnitude effects for the purposes of the present application, as stated in Appendix G.

Table 3.

Numerical values employed to calculate the unsteady indicator function.

Through excluding the change in the lift coefficients of the fixed wings with different local sweep angles before morphing, the evaluated sweeping-modification lift coefficient is written as

(3.19) \begin{equation} {{\hat C}_{L, \Lambda }}\left ( {t} \right )={{{\hat C}^{\prime \prime}}_{L, \Lambda }}\left ( {t} \right )-{{{\hat C}^{\prime \prime}}_{L, \Lambda }}\left ( t_0 \right ), \end{equation}

where the excluded initial lift coefficient ${{\hat C}^{\prime \prime}}_{L, \Lambda } ( t_0 )$ is

(3.20) \begin{aligned} {{\hat C}^{\prime \prime}}_{L, \Lambda } (t_0) &= \frac {1}{2} \frac {{{\kappa }_{{2\text{-}\textrm{D}}}}}{\displaystyle {{E}_{ \Lambda }}(t_0)+\frac {{{\kappa }_{{2\text{-}\textrm{D}}}}}{{\pi} {{AR}}}{{ \left [ \frac {{{U}_{\infty }}}{{{U}_{n}}(t_0)} \right ]}^{2}} \left [\frac {1+{\delta }_{ \Lambda }(t_0)}{\sqrt {1+k_T^2}}\right ]} \sin (2 \alpha ) \\ &= \frac {1}{2} \frac {{{\kappa }_{{2\text{-}\textrm{D}}}}}{\displaystyle {\sqrt {\frac {1}{{{\cos }^{2}} \Lambda _{e,0}}+{{\left ( \frac {{{\kappa }_{{2\text{-}\textrm{D}}}}}{{\pi} {{AR}}} \right )}^{2}}}}+\frac {{{\kappa }_{2{\text{-}\textrm{D}}}}}{{\pi} {{AR}}}{{ \left [ \frac {{{U}_{\infty }}}{{{{U}_{x}}\left ( t_0 \right )\cos {{ \Lambda }_{e,0}}}} \right ]}^{2}}\left [\frac {1+{\delta }_{ \Lambda }(t_0)}{\sqrt {1+k_T^2}}\right ]} \sin (2 \alpha ) \\ &= \frac {1}{2} \frac {{{\kappa }_{{2\text{-}\textrm{D}}}}\;{{\cos }^{2}}{{ \Lambda }_{e,0}}}{\displaystyle {{\cos }^{2}}{{ \Lambda }_{e,0}}\sqrt {\frac {1}{{{\cos }^{2}}{{ \Lambda }_{e,0}}}+{{\left ( \frac {{{\kappa }_{{2\text{-}\textrm{D}}}}}{{\pi} {{AR}}} \right )}^{2}}}+\frac {{{\kappa }_{{2\text{-}\textrm{D}}}}}{{\pi} {{AR}}}\left [\frac {1+{C_{\delta 1}} \left ( { \Lambda }_{e,0}-{C_{\delta 2}} \right )^2 + {C_{\delta 3}}}{\sqrt {1+k_T^2}}\right ]} \sin \left ( 2\alpha \right ). \end{aligned}

In this derivation, the relation ${{U}_{x}}(t_0)={U}_{\infty }$ is applied.

Dividing the evaluated sweeping-modification lift coefficient (3.19) by its maximum, we get the evaluated scaled sweeping-modification lift coefficient

(3.21) \begin{equation} {\hat C}_{L, \Lambda }^{*}\left ( t \right )=\frac {{{\hat C}_{L, \Lambda }}\left ( t \right )}{{{\hat C}_{L, \Lambda , {max}}}}. \end{equation}

Similarly, in the backward sweeping, the evaluated scaled sweeping-modification lift coefficient is

(3.22) \begin{equation} {\hat C}_{L, \Lambda }^{*}\left ( t \right )=\frac {{{\hat C}_{L, \Lambda }}\left ( t \right )}{{{\hat C}_{L, \Lambda , {min}}}}. \end{equation}

As shown in figure 18, the evaluated scaled sweeping-modification lift coefficient reflects the changes in the normal velocity components $U_n$ in figure 17, especially the symmetry around the mid-moment. This symmetry originates from the horizontal component of the morphing velocity, $U_{s}\cos \alpha$ . The symmetric component forms the basis of the sweeping-modification lift coefficient. Moreover, the evaluated scaled sweeping-modification lift coefficients under the forward and backward sweeping cases exhibit consistent trends to their numerical counterparts in figures 15 and 28. For the forward sweeping, the theoretical model captures the similarity of the numerical results during the ascending stage, ${ ( t-{{t}_{0}} )}/{ ( {{t}_{f}}-{{t}_{0}} )}\in [ 0,0.5 ]$ . For the backward sweeping, it also predicts an increase when ${ ( t-{{t}_{0}} )}/{ ( {{t}_{f}}-{{t}_{0}} )}\in [ 0.15,0.45 ]$ and a decrease near the mid-moment as the morphing duration is prolonged. These consistencies indicate that

(3.23) \begin{equation} {\hat C}_{L, \Lambda }^{*} = { C}_{L, \Lambda }^{*} \end{equation}

or

(3.24) \begin{equation} \frac {{{\hat C}_{L, \Lambda }}\left ( t \right )}{{{\hat C}_{L, \Lambda , {max/min}}}}= \frac {{{C}_{L, \Lambda }}\left ( t \right )}{{{C}_{L, \Lambda , {max/min}}}}. \end{equation}

Here, the consistent similarity results between the numerical simulation and theoretical model provide a means for further correcting the magnitude of the maximum lift over/undershoot. With the consistent similarity, the lift model is augmented by including the viscous effects associated with flow separation. The possible underestimation in lift due to additional nonlinearity of the lift curve in the pre-stall regions (Hewes Reference Hewes1978) can be compensated for.

Figure 18.

Scaled sweeping-modification lift coefficient evaluated with the proposed model: (a) forward sweeping; (b) backward sweeping.

From (3.24), the sweeping-modification lift coefficient is formulated as

(3.25) \begin{equation} {{{C}_{L, \Lambda }}\left ( t \right )} = {{{\hat C}_{L, \Lambda }}\left ( t \right )}\; \frac {{{C}_{L, \Lambda , {max/min}}}}{{{\hat C}_{L, \Lambda , {max/min}}}}. \end{equation}

This expression, along with (3.1), (3.2) and (3.16)–(3.20), constitutes the reduced-order unsteady model for the lift over/undershoot from the local sweep morphing, which can be used to evaluate the total lift coefficient during the morphing by adding the pre-morphing lift coefficient.

The proposed lift model is reduced-order since it captures the essential dynamics in lift generation during the local sweep morphing with a relatively small number of physical factors (Lucia, Beran & Silva Reference Lucia, Beran and Silva2004; Karbasian & Vermeire Reference Karbasian and Vermeire2022). The model provides quantitatively accurate characterisations of the local sweep morphing dynamics at a computational cost substantially lower than that of the original numerical model, while offering an accessible framework for readily interpreting the underlying dynamics. In this model, the dominant factors considered are the horizontal morphing velocity, non-elliptic circulation distribution and spanwise flow induced by the local sweep angles, high geometric angle of attack and unsteady wake memory effect.

Figure 19 presents the prediction of the reduced-order unsteady model and the result of the numerical simulation for the sweeping-modification lift coefficient when the change in the morphing start is considered. The numerical result is the average sweeping-modification lift coefficient under the four typical morphing starts and the same morphing duration of $2T$ . The theoretical prediction is in reasonable agreement with the numerical result. In figure 19(a), the reduced-order unsteady model predicts the increase in the sweeping-modification lift coefficient during the accelerating stage $ ( t - {{t}_{0}} )/T\in [ 0,1 ]$ of the forward sweeping (that is, the ascending stage of the sweeping-modification lift coefficient). Nevertheless, the reduced-order unsteady model underestimates the maximum value of the sweeping-modification lift coefficient. Compared with the prediction of the reduced-order unsteady model, the numerical results are characterised by an earlier timing of the maximum sweeping-modification lift coefficient and a rapid decay after the maximum value, which determine the primary and secondary peaks of the total lift coefficient, respectively. The prediction for backward sweeping is similar to that for forward sweeping, as shown in figure 19(b). The difference between the theoretical and numerical sweeping-modification lift coefficients arises from the additional unsteady effects caused by the local sweep morphing. These unsteady effects lead to the asymmetry in the profile of the sweeping-modification lift coefficient during the ascending stage $ ( t - {{t}_{0}} )/T\in [ 0,1 ]$ and descending stage $ ( t - {{t}_{0}} )/T\in [ 1,2 ]$ .

Figure 19.

Comparison between the theoretical prediction and the numerical simulation for the sweeping-modification lift coefficient when the morphing start changes: (a) forward sweeping; (b) backward sweeping. The solid line corresponds to the theoretical model, while the dashed line represents the average value of the numerical results under the four typical morphing starts in figure 12.

Figures 20 and 21 show the comparison between the reduced-order unsteady model and the numerical simulation for the sweeping-modification lift coefficients under the morphing durations of $T$ , $1.5T$ , $2T$ , $2.5T$ and $3T$ . The morphing start is fixed at the valley moment for the forward sweeping and the peak moment for the backward sweeping. Overall, the predictions of the reduced-order unsteady model present a good alignment with the numerical results, although there are apparent deviations in the slow sweeping mode. The prediction error is measured by the time-averaged absolute value of the difference between the theoretical and numerical sweeping-modification lift coefficients, that is,

(3.26) \begin{equation} e=\frac {1}{{{t}_{f}}-{{t}_{0}}}\int _{{{t}_{0}}}^{{{t}_{f}}}{\left | C_{L, \Lambda }^{({theory})}\left ( t \right )-C_{L, \Lambda }^{({num})}\left ( t \right ) \right |}\,{\textrm{d}}t. \end{equation}

Figure 20.

Comparison between the theoretical prediction and the numerical simulation for the sweeping-modification lift coefficient of the forward sweeping when the morphing duration changes: (a) fast sweeping mode; (b) slow sweeping mode. The solid lines correspond to the theoretical model, while the dashed lines represent numerical results.

Figure 21.

Comparison between the theoretical prediction and the numerical simulation for the sweeping-modification lift coefficient of the backward sweeping when the morphing duration changes: (a) fast sweeping mode; (b) slow sweeping mode. The solid lines correspond to the theoretical model, while the dashed lines represent numerical results.

For the forward sweeping cases, the prediction error of the model is almost less than 0.06, except for the maximum error of 0.09 observed at the morphing duration of $T$ , as exhibited in figure 22( a). The prediction error for backward sweeping cases is relatively uniform, with a maximum error of only approximately 0.03, which corresponds to the morphing duration of $2.25T$ , as shown in figure 22( b). Discrepancies in the sweeping-modification lift coefficient emerge during the deceleration stage under the fast morphing mode and increase under the slow morphing mode. At longer morphing durations, the low horizontal morphing velocity weakens the role of this factor, including the associated higher-order unsteady effects, and brings out other inherent higher-order unsteady effects in the flow.

Figure 22.

Error between the theoretical prediction and the numerical simulation for the sweeping-modification lift coefficient across different morphing durations: (a) forward sweeping; (b) backward sweeping.

The contribution of unsteady effects attributed to the local sweep morphing can be quantified through the numerical results and theoretical formula (3.16). The unsteady effects comprise low-order components, characterised by (3.17), and higher-order components. Rearranging the expression (3.16) yields

(3.27) \begin{equation} \varPhi \left ( t \right )=\frac {1+{{\delta }_{ \Lambda }}\left ( t \right )}{\displaystyle \frac {\pi AR}{{{\kappa }_{\text{2-}\textrm{D}}}}\left [ \frac {{{\kappa }_{\text{2-}\textrm{D}}}\sin \left ( 2\alpha \right )}{2{{{{C}^{\prime \prime}}}_{L, \Lambda }}\left ( t \right )}-{{E}_{ \Lambda }}\left ( t \right ) \right ]{{\left [ \frac {{{U}_{n}}\left ( t \right )}{{{U}_{\infty }}} \right ]}^{2}}}. \end{equation}

A sweeping indicator can be defined in an analogous way to the sweeping-modification lift coefficient. The lift coefficient induced by local sweep morphing and extended pre-morphing lift coefficient are substituted into (3.27) to obtain the corresponding unsteady indicator functions $\varPhi$ . Their difference $\Delta \varPhi$ is first time-averaged and its absolute value is then taken to ensure positive values under the backward sweeping cases. The sweeping indicator is formulated as

(3.28) \begin{equation} {{\varPhi }_{ \Lambda }}=\frac {1}{{{t}_{f}}-{{t}_{0}}}\left | \int _{{{t}_{0}}}^{{{t}_{f}}}{\Delta \varPhi \left ( t \right )}\;{ }{\textrm d}t \right |, \end{equation}

which measures the intensity of the unsteady effects associated with the horizontal morphing velocity.

In the slow mode of forward sweeping, the sweep indicator ${\varPhi }_{ \Lambda }$ decreases with the morphing duration, as shown in figure 23. This indicates that a reduction in the horizontal morphing velocity is accompanied by weaker higher-order unsteady effects associated with it. For the backward sweeping, the sweep indicator ${\varPhi }_{ \Lambda }$ exhibits almost no change with the morphing duration (remaining around 0.014), which reflects that the higher-order unsteady effects associated with the horizontal morphing velocity are insensitive to the morphing duration. At the morphing duration of $4T$ , the sweeping indicators for the forward sweeping and backward sweeping cases reach a near-identical level.

Figure 23.

Sweeping indicator across different morphing durations. The square symbol represents the forward sweeping, while the diamond symbol denotes the backward sweeping.

Moreover, the added-mass lift component is neglectable in the modelling because the adopted wing model has zero thickness and its local sweep morphing is limited to the wing plane. For an accelerating elliptic aerofoil immersed in a constant free stream, Limacher (Reference Limacher2021) decomposes the total added-mass force into a chordwise component ${{F}_{a,\shortparallel }}=- ( {\pi }/{4}\; )\rho h^{2}a_{\shortparallel }$ and a chord-normal component ${{F}_{a,\bot }}=- ( {\pi }/{4}\; )\rho c^{2}a_{\bot }$ , where $h$ denotes aerofoil thickness (minor axis length), $c$ denotes aerofoil chord (major axis length), $a_{\shortparallel }$ is the chordwise acceleration component and $a_{\bot }$ is the chord-normal acceleration component. In this work, the conditions of the zero thickness wing and the in-plane morphing are equivalent to $h=0$ and $a_{\bot }=0$ , respectively. Thus, both the two added-mass force components vanish. This estimation still holds when three-dimensional effects are considered (Brennen Reference Brennen1982; Manar & Jones Reference Manar and Jones2019). Even for wing accelerations where the translating direction forms a large angle with the wing plane, the added-mass lift component accounts for only a small fraction of the total lift in both fast and slow modes (Mancini et al. Reference Mancini, Manar, Granlund, Ol and Jones2015).

3.5. Vortex dynamics of the local sweep morphing

The asymmetry in the sweeping-modification lift coefficient profile correlates with the vortex dynamics of the local sweep morphing. This correlation is revealed by examining how the local sweep morphing modifies the vortex dynamics of the pre-morphing configuration.

Before the forward sweeping, the maximum average lift coefficient configuration generates the highly unsteady flow, which is characterised by the alternating formation, growth and shedding of the LEVs and TEVs. Figure 24 displays the instantaneous vortex structures at two typical moments corresponding to the lift peak and the lift valley. The vortex structures of the maximum average lift coefficient configuration are identified using the $Q$ -criterion. The isosurfaces of $Q=2.5$ are plotted and coloured by the normalised spanwise vorticity ${\omega }_y$ . The distribution of the spanwise vorticity within a slice at the spanwise position ${y}/({b}_{\textit{iw}}+{b}_{\textit{ow}})=37\,\%$ is presented in the right-hand side of each panel. The slice is represented by the solid black line on the wing surface in the left-hand side. As the lift coefficient ascends and approaches its peak, the streamwise size of the LEVs grows continuously. Concurrently, the TEVs are convected downstream until they shed. At the moment of the lift peak, the streamwise size of the LEVs is close to the local chord length of the wing section, as shown in figure 24(a). The LEVs upstream interact with the detached TEVs downstream, producing a downwash flow between them. In the subsequent descending of lift coefficient, new TEVs start to form and the spiral structure of the LEVs is broken. At the moment of the lift valley, the TEVs upstream interact with the detached LEVs downstream, inducing an upwash flow between them, as shown in figure 24(b). The near-field vorticity carries significant weight in the lift generation of the fixed-wing, as demonstrated in the force decomposition theory (Chang Reference Chang1992; Chang, Yang & Chu Reference Chang, Yang and Chu2008). In the wing section, the near-wake is dominated only by the large-scale LEVs and TEVs, without smaller-scale secondary vortices. Consequently, the lift coefficients exhibits no high-frequency components in figure 6.

Figure 24.

Instantaneous vortices of the maximum average lift coefficient configuration: (a) moment of the lift peak; (b) moment of the lift valley. In the left-hand side of each panel, the vortex structures of the fixed wing with different local sweep angles are identified using the isosurfaces of $Q = 2.5$ and coloured by the normalised spanwise vorticity ${\omega }_y$ . The right-hand side shows the spanwise vorticity within a slice at the spanwise position ${y}/({b}_{\textit{iw}}+{b}_{\textit{ow}})=37\,\%$ , which is indicated by the solid black line on the wing surface in the left-hand side.

To explain the asymmetry of the sweeping-modification lift coefficient profile, we select the mid-moments, ${(t-{{t}_{0}})}/{T}=0.5$ and ${(t-{{t}_{0}})}/{T}=1.5$ , of the forward-sweeping acceleration and deceleration stages in figure 25, corresponding to the two peak values (see figure 11 a) of the extended pre-morphing lift coefficient, respectively. At these two moments, the relative streamwise position between the LEVs and TEVs is consistent with that at the peak moments of the pre-morphing lift coefficient. That is, the LEVs are located upstream of the detached TEVs. However, the normal distances of the LEVs from the upper surface of the wing are different. This is closely related to the change in the vertical velocity component, ${{U}_{s}}\sin \alpha$ (see figure 16), caused by the local sweep morphing. When the vertical morphing velocity increases, the LEVs approach the upper wing surface. Conversely, when the vertical morphing velocity decreases, the LEVs move farther from the upper wing surface. The vertical morphing velocity changes the interaction between the LEVs and the upper wing surface, leading to the asymmetric profile of the sweeping-modification lift coefficient.

Figure 25.

Instantaneous vortices of the morphing wing during the forward sweeping: (a) mid-moment ${(t-{{t}_{0}})}/{T}=0.5$ of the acceleration stage; (b) mid-moment ${(t-{{t}_{0}})}/{T}=1.5$ of the deceleration stage. The forward sweeping is characterised by the nonlinear sine morphing kinematics with the morphing start of the valley moment ( $t_v$ ) and the morphing duration of $2T$ . In the left-hand side of each panel, the vortex structures of the local sweeping wing are identified using the isosurfaces of $Q = 2.5$ and coloured by the normalised spanwise vorticity ${\omega }_y$ . The slice in the right-hand side has the same spanwise position as that in figure 24.

Figure 9 shows that at the morphing durations of $3T$ and $4T$ , the second lift peak becomes larger than the first, which is different from the trend observed at shorter morphing duration of $2T$ . This phenomenon is related to the vortex formation and interaction modulated by the horizontal morphing velocity. The decomposition of the lift coefficient facilitates the extraction of key vortex dynamics. The sweeping-modification lift coefficient reflects the vortex dynamics associated with the horizontal morphing velocity. The symmetric component of the sweeping-modification lift coefficient reaches its maximum at the mid-moment of the morphing phase. For the morphing duration of $3T$ , the second one among the three peaks of the extended pre-morphing lift coefficient during the morphing phase aligns with the peak of the sweeping-modification lift coefficient. It is obvious that the second lift peak becomes the largest after superimposition. For the morphing duration of $4T$ , the extended pre-morphing lift coefficient exhibits two peaks respectively before and after the mid-moment. Since the second lift peak is located in the acceleration phase with the vortex lift advantage (as detailed previously) and is closer to the moment of maximum horizontal morphing velocity than the first peak, it generates the highest lift.