3.1. Force evolution across different ${\textit{Re}}$ and ${\textit{EI}}$

The transient hydrodynamic force coefficients for circular plates with three orders of magnitude in bending stiffness, ${\textit{EI}} = 1.72\, \mathrm{Nm}$ , $5.79\times 10^{-3} \, \mathrm{Nm}$ and $4.19\times 10^{-4} \, \mathrm{Nm}$ , under varying Reynolds numbers ${\textit{Re}}$ are shown in figure 6.

For the plate with bending stiffness ${\textit{EI}} = 1.72\, \mathrm{Nm}$ (corresponding to a thickness of $h = 2.00 \, \mathrm{mm}$ ), which behaves as a rigid plate with negligible deformation, the force curves collapse well across different Reynolds numbers, confirming a clear Reynolds number independence (Fernando & Rival Reference Fernando and Rival2016b ; Li et al. Reference Li, Xiang, Qin, Liu and Wang2022). Furthermore, the force evolution exhibits a distinct peak-and-valley pattern (figure 6 a). During acceleration, the force increases rapidly with $t^*$ , reaching a peak at $t^* = 2$ , followed by a decrease to a minimum at $t^* \approx 5.5$ – the so-called ‘drag trough’ (Fernando & Rival Reference Fernando and Rival2016a ) – before rising again.

As the bending stiffness decreases to ${\textit{EI}} = 5.79\times 10^{-3} \, \mathrm{Nm}$ (corresponding to a plate thickness of $h = 0.30 \, \mathrm{mm}$ ), the force response exhibits significant changes with increasing Reynolds number, leading to a breakdown of Reynolds number independence (figure 6 b). At ${\textit{Re}} = 4 \times 10^4$ , the characteristic peak-and-valley pattern persists, with a distinct force peak at $t^* = 2$ and a trough at $t^* \approx 5.5$ . However, compared with the rigid plate, the peak force is reduced, while the trough force is elevated. As ${\textit{Re}}$ increases, the force peak gradually flattens, and the trough becomes less pronounced. At ${\textit{Re}} = 10 \times 10^4$ , the peak-and-valley pattern disappears entirely, resulting in a smooth and uniform force evolution with no discernible extrema. Notably, the force coefficient at $t^* \approx 5.5$ initially increases with ${\textit{Re}}$ before declining at higher values. These results indicate that, as the bending stiffness decreases, the unsteady force dynamics lose their Reynolds number independence, underscoring the coupled effects of flexibility and Reynolds number on force evolution.

For plates with further reduced bending stiffness to ${\textit{EI}} = 4.19\times 10^{-4} \, \mathrm{Nm}$ (corresponding to a thickness of $h = 0.125 \, \mathrm{mm}$ ), the force curves remain non-collapsing across Reynolds numbers but exhibit consistent trends: a rapid rise during acceleration followed by stabilization with negligible fluctuations. The peak-and-valley pattern disappears entirely.

Overall, these results emphasize the coupled effects of Reynolds number and flexibility on unsteady force dynamics, causing the force evolution on reconfigurable plates to deviate from the Reynolds number independence and peak-and-valley pattern observed in rigid plates.

3.2. Non-dimensional bending stiffness ${\textit{EI}}^*$ independence of force evolution

The breakdown of Reynolds number scaling stems from its primary focus on fluid dynamics, whereas the reconfigurable plate in this study undergoes deformation that modifies the surrounding flow, resulting in a coupled fluid–structure interaction. To characterize this interaction, a non-dimensional bending stiffness, ${\textit{EI}}^*$ , is employed as

(3.1) \begin{equation} {\textit{EI}}^* = \frac {\textit{EI}}{\rho _{\!f} U_c^2 D^3}. \end{equation}

The physical significance of ${\textit{EI}}^*$ has been interpreted differently across various studies. From an energy perspective, Schouveiler & Boudaoud (Reference Schouveiler and Boudaoud2006) defined the elastohydrodynamical number, expressed as $ 1/{\textit{EI}}^*$ , to represent the ratio of flow-induced potential energy to elastic bending energy. Similarly, Alben et al. (Reference Alben, Shelley and Zhang2002, Reference Alben, Shelley and Zhang2004) introduced a parameter equivalent to $ \sqrt {1/{\textit{EI}}^*}$ , balancing fluid kinetic energy with elastic potential energy. Alternatively, from a force perspective, De Langre (Reference De Langre2008) and Gosselin et al. (Reference Gosselin, De Langre and Machado-Almeida2010) considered the Cauchy number, $Ca$ , expressed as $ 1/{\textit{EI}}^*$ , as the ratio of fluid dynamic pressure to plate bending rigidity. In the present study, ${\textit{EI}}^*$ is adopted from the perspective of wing flapping aerodynamics (Shyy et al. Reference Shyy, Aono, Chimakurthi, Trizila, Kang, Cesnik and Liu2010; Kang et al. Reference Kang, Aono, Cesnik and Shyy2011), characterizing the balance between elastic bending forces and aerodynamic forces. The derivation of ${\textit{EI}}^*$ can be found in the Supplementary material.

Three orders of magnitude of ${\textit{EI}}^*$ were selected as representative cases (table 3). The corresponding deformation and force data are shown in figure 7. From figure 7, despite variations in ${\textit{EI}}$ and $ Re$ , the deformation ( $ S_p^*$ defined in (2.7)) and force evolution ( $ C_H$ defined in (2.11) and $ C_H^*$ defined in (2.13)) collapse onto a single curve as a function of formation time, with only limited deviations under the same ${\textit{EI}}^*$ condition.

The deformation behaviour under different ${\textit{EI}}^*$ values is presented in figure 7(a,b). For ${\textit{EI}}^* = 2.66 \times 10^{-1}$ , no deformation is observed, and the plate maintains its projected area as a perfect circle, effectively behaving as rigid. At ${\textit{EI}}^* = 1.16 \times 10^{-2}$ , moderate deformation occurs, categorizing the plate as flexible. For ${\textit{EI}}^* = 1.19 \times 10^{-2}$ , significant deformation reduces the projected area to nearly half its original size, showing the plate as extra flexible.

As for the force evolution, from figure 7(d,g), when ${\textit{EI}}^* = 2.66 \times 10^{-1}$ , the instantaneous force evolution follows the so called peak-and-valley pattern as discussed in § 3.1. When ${\textit{EI}}^*$ decreases to $ 1.16 \times 10^{-2}$ (figure 7 e,h), the peak-and-valley characteristic weakens: the peak force decreases, the valley force increases and the force stabilizes around the steady drag coefficient after $ t^* \approx 4$ . Further reduction of ${\textit{EI}}^*$ to $ 1.19 \times 10^{-2}$ eliminates the peak-valley pattern entirely (figure 7 f,i). The instantaneous force increases during the acceleration phase and stabilizes at $ t^* \approx 1.5$ , earlier than the end of acceleration ( $ t^* = 2$ ). However, regardless of whether $ C_H$ or $ C_H^*$ is considered, the instantaneous force coefficient curves collapse under identical ${\textit{EI}}^*$ . This demonstrates that force evolution is primarily governed by ${\textit{EI}}^*$ and remains consistent under the same non-dimensional bending stiffness conditions.

Figure 7.

Instantaneous $ S_p^*$ (a–c), $ C_H$ (d–f) and $ C_H^*$ (g–i) for three orders of magnitude of ${\textit{EI}}^*$ ; legend details can be found in table 3.

3.3. Load-shifting phenomenon in force evolution

To further investigate the impact of different values of ${\textit{EI}}^*$ on the force evolution pattern, the reconfiguration number, originally defined under steady conditions by Gosselin et al. (Reference Gosselin, De Langre and Machado-Almeida2010), is extended to the unsteady accelerating process. The reconfiguration number $ R(t^*)$ is defined as

(3.2) \begin{equation} R(t^*) = \frac {C_{H_{\!f}}(t^*)}{C_{H_r}(t^*)}, \end{equation}

where $ C_{H_{\!f}}(t^*)$ and $ C_{H_r}(t^*)$ represent the instantaneous hydrodynamic force coefficients of the flexible/extra-flexible and rigid plates, respectively. This normalization employs the rigid response as a dynamic baseline, thereby removing the inherent time-dependent variations of the accelerating flow and isolating the effect of reconfiguration (Gosselin et al. Reference Gosselin, De Langre and Machado-Almeida2010). A value of $R(t^*) \lt 1$ indicates force reduction relative to the rigid case, and $R(t^*) \gt 1$ indicates enhancement.

Figure 8.

The reconfiguration number $ R$ and the absolute hydrodynamic force coefficient difference $ \Delta C_H$ for the flexible plate (a,b) and extra-flexible plate (c,d). Legend details can be found in table 3.

Additionally, to clearly present the results, the absolute hydrodynamic force coefficient difference $ \Delta C_H$ is defined as

(3.3) \begin{equation} \Delta C_H(t^*) = C_{H_{\!f}}(t^*) - C_{H_r}(t^*). \end{equation}

A negative value ( $\Delta C_H \lt 0$ ) indicates force reduction, whereas a positive value ( $\Delta C_H \gt 0$ ) represents force enhancement. While $R(t^*)$ is ideal for assessing the relative impact of reconfiguration, $\Delta C_H(t^*)$ provides a direct measure of the force changes.

Results for $ R$ and $ \Delta C_H$ for flexible plates with ${\textit{EI}}^* = 1.16 \times 10^{-2}$ and extra-flexible plates with ${\textit{EI}}^* = 1.19 \times 10^{-2}$ are shown in figure 8.

As shown in figure 8(a,b), during acceleration, the force on the flexible plate is lower than that on the rigid plate ( $ R \lt 1$ and $ \Delta C_H \lt 0$ ), with the difference increasing over time and peaking at $ t^* = 2$ (end of acceleration). Subsequently, the difference diminishes, and the forces equalize at $ t^* \approx 2.8$ ( $ R = 1$ and $ \Delta C_H = 0$ ). Beyond this point, the force on the flexible plate exceeds that on the rigid plate ( $ R \gt 1$ and $ \Delta C_H \gt 0$ ), with the maximum difference observed at $ t^* = 5.5$ . This pattern of force evolution is herein referred to as ‘load-shifting’. For the extra-flexible plate (figure 8 c,d), the hydrodynamic force coefficient remains consistently lower than that of the rigid plate.

Based on these results, the classification of rigid, flexible and extra-flexible plates can be more clearly defined. A rigid plate is characterized by its force evolution following Reynolds number independence. In contrast, flexible and extra-flexible plates exhibit Reynolds-number-dependent force evolution during acceleration. A flexible plate exhibits load-shifting phenomenon, with its peak force at $ t^* = 2$ being lower than that of the rigid plate and its valley force at $ t^* = 5.5$ being higher. An extra-flexible plate, however, does not exhibit load-shifting phenomenon, as its force evolution remains consistently lower than that of the rigid plate throughout the acceleration process.

Reynolds number scaling is well recognized (Fernando & Rival Reference Fernando and Rival2016a , Reference Fernando and Rivalb ; Grift et al. Reference Grift, Vijayaragavan, Tummers and Westerweel2019; Li et al. Reference Li, Xiang, Qin, Liu and Wang2022, Reference Li, Chen, Xiang, Liu and Wang2024), while load-shifting phenomenon is particularly relevant in biological propulsion, as it indicates that reconfiguration can reduce force variation without altering the mean force (Shyy et al. Reference Shyy, Aono, Chimakurthi, Trizila, Kang, Cesnik and Liu2010; Kim & Gharib Reference Kim and Gharib2011a ). From a gust alleviation perspective, reconfiguration can mitigate instantaneous high loads and reduce force fluctuations, improving biological flight stability and enhancing vegetation survival (Triantafyllou et al. Reference Triantafyllou, Weymouth and Miao2016; Cheney et al. Reference Cheney, Stevenson, Durston, Song, Usherwood, Bomphrey and Windsor2020). Extended discussions on the physical value of load-shifting can be found in the Supplementary material.

Furthermore, when ${\textit{EI}}^*$ is large (figure 7 a,d,g), the plate behaves rigidly without deformation. Conversely, when ${\textit{EI}}^*$ is too small, the plate collapses entirely (figure 7 c,f,i), and the instantaneous force remains consistently below the minimum force of the rigid plate (figure 8 c,d). This indicates that both excessively large and small ${\textit{EI}}^*$ are suboptimal.

Figure 9.

Instantaneous force coefficients at $ t^* = 2$ (a) and $ t^* = 5.5$ (b) against non-dimensional bending stiffness ${\textit{EI}}^*$ (or, the Cauchy number, $Ca=1/{\textit{EI}}^*$ ), for different bending stiffness ${\textit{EI}}$ as listed in table 2 and four Reynolds numbers $ Re = 4.0, 6.0, 8.0, 10.0 \times 10^4$ . The boundaries (vertical light grey dashed lines) between extra-flexibility, flexibility and rigidity are identified at two inflection points: ${\textit{EI}}^* = 2.28 \times 10^{-3}$ and ${\textit{EI}}^* = 0.143$ . The thick grey solid line is the power-law fit from (3.10).

3.4. Load-shifting regime boundaries defined by ${\textit{EI}}^*$

To determine the boundaries of this load-shifting phenomenon, the instantaneous forces were measured for each plate ( ${\textit{EI}}$ ranging from $ 4.19 \times 10^{-4}$ to $ 36.2$ Nm) in table 2 at Reynolds numbers $ Re = 4.0, 6.0, 8.0, 10.0 \times 10^4$ , covering an ${\textit{EI}}^*$ range from approximately $ 10^{-4}$ to $ 10^2$ . Force coefficients at $ t^* = 2$ (peak) and $ t^* = 5.5$ (valley) were extracted and plotted against ${\textit{EI}}^*$ (figure 9). Variations of the reconfiguration number $ R$ and force difference $ \Delta C_H$ , referenced to the rigid plate at $ t^* = 2$ and $ t^* = 5.5$ , are shown in figure 10. As ${\textit{EI}}^*$ decreases, both $ R$ and $ \Delta C_H$ at $ t^* = 2$ decline, whereas at $ t^* = 5.5$ , they increase and then decrease.

Figure 10.

Absolute hydrodynamic force coefficient difference $ \Delta C_H$ (a,b) as functions of ${\textit{EI}}^*$ . Panels (a) and (c) correspond to the results at $ t^* = 2$ , while panels (b) and (d) correspond to the results at $ t^* = 5.5$ . Here $ \Delta C_H$ is referenced to the rigid plate values, as detailed in figure 9. The vertical light grey dashed lines indicate the boundaries between extra-flexible, flexible and rigid, identified at ${\textit{EI}}^* = 2.28 \times 10^{-3}$ and ${\textit{EI}}^* = 0.143$ .

The load-shifting phenomenon is defined by a peak force reduction ( $R(t^*=2) \lt 1$ ) coupled with a valley force enhancement ( $R(t^*=5.5) \gt 1$ ). As neither the rigid ( $R \approx 1$ ) nor the extra-flexible ( $R(t^*=5.5) \lt 1$ ) plates satisfy these conditions, the load-shifting phenomenon is confined to the intermediate ${\textit{EI}}^*$ regime. The following analysis seeks to identify the critical transition points, ${\textit{EI}}_{{{\textit{crit}},1}}^{*}$ and ${\textit{EI}}_{{{\textit{crit}},2}}^{*}$ , that delineate the boundaries of this flexible regime.

The flexible-to-rigid transition ( ${\textit{EI}}_{{{\textit{crit}},2}}^{*}$ ). This occurs when the plate’s elastic restoring moment can no longer balance the peak hydrodynamic bending moment, marking the onset of deformation. Since the plate undergoes symmetric beam-like bending (figure 3), the peak hydrodynamic moment can be written as

(3.4) \begin{equation} M_{{{\textit{hydro}},{\textit{peak}}}} = F_{H,{\textit{half}}}\,{\times}\,L_{{\textit{arm}}}, \end{equation}

where $F_{H,{\textit{half}}} = F_{H,{{\textit{peak}}}} /2$ is the hydrodynamic force on half of the plate, and $L_{{\textit{arm}}} = k_{p}(D/2)$ is the effective moment arm with $k_{p}$ denoting a pressure distribution factor (Batchelor Reference Batchelor2000).

The elastic resisting moment of the deforming half-plate, $M_{\textit{resist}}$ , follows from plate theory (Landau et al. Reference Landau, Pitaevskii, Kosevich and Lifshitz2012). The moment per unit length scales as $M_l \sim EI/D$ , and integrating over the deforming radius ( $D/2$ ) gives

(3.5) \begin{equation} M_{\textit{resist}} \approx M_l {\times}D/2 \approx EI/2. \end{equation}

Equating $ M_{{{\textit{hydro}},{\textit{peak}}}} \approx M_{\textit{resist}}$ yields the predicted critical stiffness:

(3.6) \begin{equation} EI_{{{\textit{crit}},2}}^{*} = \frac {\textit{EI}}{\rho _{f}U_{c}^{2}D^{3}} \approx \frac {\pi }{16}k_{p}C_{H,{{\textit{rigid}},{\textit{peak}}}}. \end{equation}

Here, $C_{H,{{\textit{rigid}},{\textit{peak}}}}$ denotes the peak force coefficient of a rigid plate, which is primarily governed by $a^*$ (Li et al. Reference Li, Xiang, Qin, Liu and Wang2022, Reference Li, Chen, Xiang, Liu and Wang2024), ${\textit{Re}}$ (Liu & Sun Reference Liu and Sun2024) and the plate geometry (Fernando & Rival Reference Fernando and Rival2016a , Reference Fernando and Rivalb ; Reijtenbagh et al. Reference Reijtenbagh, Tummers and Westerweel2023a ), while additional factors may also contribute. In this study for $a^*=0.25$ , ${\textit{Re}}\sim 10^4$ and circular plates, $C_{H,{{\textit{rigid}},{\textit{peak}}}} \approx 1.73$ (figure 9). And, assuming a uniform pressure distribution gives $k_p \approx 0.42$ . Substituting into (3.6) yields

(3.7) \begin{equation} EI_{{{\textit{crit}},2}}^{*} \approx 0.143. \end{equation}

This theoretical prediction ( ${\approx} 0.143$ ) agrees reasonably with the value reported by Baskaran et al. (Reference Baskaran, Hutin and Mulleners2023) ( ${\approx} 0.111$ ). The slightly larger value in the present study can be attributed to two main factors. First, the accelerating flow conditions here differ from the steady flow in Baskaran et al. (Reference Baskaran, Hutin and Mulleners2023); under unsteady conditions, added-mass effects amplify the peak force (Li et al. Reference Li, Xiang, Qin, Liu and Wang2022, Reference Li, Chen, Xiang, Liu and Wang2024), thus requiring greater bending stiffness to initiate deformation and shifting the transition to a larger ${\textit{EI}}^*$ . Second, the discrepancy may also arise from the assumption of a uniform pressure distribution in our calculation. In reality, as the plate tips bend to align with the flow, the local pressure likely decreases (Gosselin et al. Reference Gosselin, De Langre and Machado-Almeida2010), shifting the centre of pressure inward and resulting in a smaller actual value for the pressure distribution factor, $k_p$ .

The extra-flexible to flexible transition ( ${\textit{EI}}_{{{\textit{crit}},1}}^{*}$ ). Once ${\textit{EI}}_{{{\textit{crit}},2}}^{*}$ is established, the forces in the extra-flexible/flexible range can be analysed with the 1/3 power law (Alben et al. Reference Alben, Shelley and Zhang2002, Reference Alben, Shelley and Zhang2004; Gosselin et al. Reference Gosselin, De Langre and Machado-Almeida2010). The distinction between extra-flexible and flexible regimes in this study is based on deviations of the force coefficients from those of a rigid plate, particularly at the force valley. From a fluid–structure interaction perspective, both regimes involve significant deformation and can therefore be described within the Vogel exponent framework (Vogel Reference Vogel1989), which underlies the 1/3 power law. In these regimes, the drag force scales with velocity as $F \propto U^{1 + \mathscr{V}}$ , with Vogel exponent $\mathscr{V}$ .

Following Gosselin et al. (Reference Gosselin, De Langre and Machado-Almeida2010) and our experimental observations in small ${\textit{EI}}^*$ regime (figures 3 and 7), two features are crucial. First, large deformations align the plate edges with the flow, minimizing their contribution to drag; only the central region near the clamped edge generates appreciable drag. Thus, the chordwise dimension $D$ becomes irrelevant as a characteristic length. Second, the plate undergoes quasi-2-D symmetric beam-like bending without 3-D folding, allowing the drag to be approximated as a 2-D force per spanwise length $F/D$ , analogous to flexible fibres in soap films (Alben et al. Reference Alben, Shelley and Zhang2002, Reference Alben, Shelley and Zhang2004).

Accordingly, the governing physical quantities are

(3.8) \begin{align} \textit{EI} \, [\mathrm{Nm}], \quad U_c \, [\mathrm{m\,s^{-1}}], \quad \rho _{\!f} \, [\mathrm{kg\,m^{-3}}], \quad F/D \, [\mathrm{N\,m^{-1}}] \end{align}.

Dimensional analysis then yields

(3.9) \begin{equation} F \sim D \rho _{f}^{2/3} U_c^{4/3} {\textit{EI}}^{1/3}. \end{equation}

This corresponds to a Vogel exponent of $\mathscr{V} = -2/3$ , consistent with the 1/3 power law proposed in Alben et al. (Reference Alben, Shelley and Zhang2002, Reference Alben, Shelley and Zhang2004), where beam bending stiffness (Beer, Johnston & DeWolf Reference Beer, Johnston and DeWolf1999) (with units N m $^2$ ) is used. Despite the unit differences, both studies conform to the same scaling exponent. This relationship can be further simplified in non-dimensional form as

(3.10) \begin{equation} C_{H} = A {\times}( {\textit{EI}}^* )^{1/3}, \end{equation}

where $A$ is derived directly from the values of $C_{H,{{\textit{rigid}},{\textit{peak}}}}$ . Since $C_{H,{{\textit{rigid}},{\textit{peak}}}}$ is influenced by $a^*$ (Li et al. Reference Li, Xiang, Qin, Liu and Wang2022, Reference Li, Chen, Xiang, Liu and Wang2024), ${\textit{Re}}$ (Liu & Sun Reference Liu and Sun2024) and plate geometry (Fernando & Rival Reference Fernando and Rival2016a , Reference Fernando and Rivalb ; Reijtenbagh et al. Reference Reijtenbagh, Tummers and Westerweel2023a ), $A$ can also be expected to depend on these physical parameters. In this study, using data from figure 9, we obtain $A = 4.56$ , with a regression fit of $R^2 = 0.928$ . The fit remains below 0.95 because, near ${\textit{EI}}^* \lesssim {\textit{EI}}^*_{{{\textit{crit}},2}}$ , the deformation is insufficient and the plate tip still contributes to the pressure distribution, violating the model assumptions. Hence, (3.10) is valid primarily below ${\textit{EI}}^*_{{{\textit{crit}},2}}$ and is not used to back-calculate ${\textit{EI}}^*_{{{\textit{crit}},2}}$ . Nevertheless, when ${\textit{EI}}^* \ll {\textit{EI}}^*_{{{\textit{crit}},2}}$ , the scaling holds. The intersection of (3.10) with the rigid trough coefficient $C_{H,{{\textit{rigid}},trough}} \approx 0.60$ gives

(3.11) \begin{equation} {\textit{EI}}^*_{{{\textit{crit}},1}} = ( 0.60/4.56 )^3 \approx 2.28 \times 10^{-3}. \end{equation}

In summary, based on moment balance and the 1/3 power law, the onset of both rigid and extra-flexible regimes is successfully predicted. The load-shifting phenomenon is confined to $2.28 \times 10^{-3} \leqslant {\textit{EI}}^* \leqslant 0.143$ , as indicated by the grey dashed lines in figures 9 and 10, where plates in the intermediate flexible regime exhibit load-shifting phenomenon.

4.1. Temporal evolution of flow fields

To investigate the temporal evolution of the flow field associated with load-shifting phenomenon, plate ‘F-7’, is taken as a representative case. The non-dimensional vorticity is used to characterize the flow field,

(4.1) \begin{equation} \omega ^* = \frac {\omega D}{U_c}, \end{equation}

where $ \omega$ is the vorticity.

Figure 11 illustrates the vortex evolution in the chordwise cross-section ( $\theta = 0^\circ$ ). The vorticity supply for vortex ring growth originates from the shear layer at the plate’s edge. As the plate moves, the fluid rolling past its edge forms a vortex ring. During the acceleration phase ( $ t^* \leqslant 2$ ), the vortex ring grows continuously. After acceleration ceases, during the constant-speed phase, the vortex ring continues to develop. At $ t^* = 3$ , the vortex ring begins to break down as vorticity is axially stretched, with portions of vorticity lagging behind the plate’s motion. At $ t^* = 4$ , the concentrated vorticity region undergoes significant breakdown into smaller patches, some of which are transported farther downstream. At $ t^* = 5$ , the breakdown intensifies, and vorticity transport in the wake becomes more pronounced. By $ t^* = 6$ , the vortex ring has completely broken down, and the vorticity fragments become highly dispersed, transitioning from an initial clustered distribution to a dispersed distribution.

In the diagonal ( $\theta = 45^\circ$ ) (figure 12) and spanwise ( $\theta = 90^\circ$ ) (figure 13) cross-sections, vorticity initially rolls up from the edges, forming a distinct vorticity layer that later rolls inward to create a concentrated vorticity region near the central axis. By $ t^* = 3$ , the vortex begins to break down, and this process becomes more pronounced as $ t^*$ increases. At $ t^* = 6$ , the vortex ring has completely broken down, and the concentrated vorticity region transitions into a dispersed state.

Figure 11.

Evolution of the non-dimensional vorticity $\omega ^*$ over non-dimensional time $t^*$ for the flexible plate ‘F-7’ in the chordwise cross-section ( $\theta = 0^\circ$ ). The thick black line indicates the deformation and position of the plate. The plate thickness is exaggerated for clarity. The dashed rectangle in panel (a) illustrates the representative integration domain used for calculating circulation (4.2) and other integral quantities ((4.3), (4.4) and (5.14)). The open circles in each panel indicate the locations corresponding to $r_v^*$ (4.3) and $z_v^*$ (4.4) within the respective measurement cross-sections.

Figure 12.

Evolution of the non-dimensional vorticity $\omega ^*$ over non-dimensional time $t^*$ for the flexible plate ‘F-7’ in the diagonal cross-section ( $\theta = 45^\circ$ ). Annotations are consistent with those in figure 11.

Another notable feature of the wake flow behind the flexible plate is its non-axisymmetry. The plate undergoes obvious bending deformation in the chordwise (figure 11) and diagonal cross-sections (figure 12), while limited deformation is observed in the spanwise cross-section (figure 13). Due to the deformation, before breakdown occurs (approximately $ t^* \lt 3$ ), the evolution of the vorticity distribution differs significantly among the three cross-sections. In the spanwise cross-section, the vorticity distribution exhibits a distinct vortex structure, whereas in the diagonal and chordwise cross-sections, apart from the vortex, a clear vorticity feeding layer is present. Axially, the vortex in the spanwise cross-section lags behind those in the other two cross-sections. Radially, the vorticity in the chordwise cross-section remains roughly parallel to the plate tip, whereas in the diagonal and spanwise cross-sections, it is more concentrated near the symmetry axis. After breakdown, this non-axisymmetry gradually diminishes. Quantitative results related to this non-axisymmetry will be discussed in § 4.2.

Figure 13.

Evolution of the non-dimensional vorticity $\omega ^*$ over non-dimensional time $t^*$ for the flexible plate ‘F-7’ in the spanwise cross-section ( $\theta = 90^\circ$ ). Annotations are consistent with those in figure 11.

Furthermore, a comparison between the flow evolution of extra-flexible, flexible and rigid plates highlights the two key features that characterize the wake of a flexible plate: non-axisymmetry and subsequent vortex breakdown, as detailed in the Supplementary materials. These features are robust; for a given dimensionless stiffness ( ${\textit{EI}}^* = 1.16 \times 10^{-2}$ ), they persist across all tested Reynolds numbers, underscoring that ${\textit{EI}}^*$ governs the fundamental flow topology.

4.2. Circulation and vorticity distribution

4.2.1. Circulation

To quantify the strength of the shed vorticity, the non-dimensional total circulation, ${\varGamma }_v^*$ , is computed by integrating the vorticity, $\omega$ , over the wake region upstream of the plate tip ( $z_{{\textit{tip}}}'$ ) as follows:

(4.2) \begin{equation} {\varGamma }_v^* = \frac {1}{U_cD} \int _0^D \int _{-D}^{z_{{\textit{tip}}}^{\prime}} \omega \, \mathrm{d}z' \, \mathrm{d}r^{\prime}. \end{equation}

The integral is formulated in a cylindrical coordinate system for consistency across the different 2-D measurement planes, where $z'$ is the streamwise axial coordinate and $r'$ is the radial coordinate. The integration domain is illustrated for a representative case in the flow fields shown in figures 11, 12 and 13. This circulation is commonly referred to as the total circulation, in contrast to vortex circulation (Gharib et al. Reference Gharib, Rambod and Shariff1998; Fernando & Rival Reference Fernando and Rival2016b ). Due to vortex breakdown, defining the boundary of individual vortex rings for calculating their circulation is challenging. Thus, the total circulation is used to evaluate the overall vorticity strength in the wake. The computed results are shown in figure 14.

The growth of circulation in different cross-sections suggests that, despite differences in non-axisymmetry, the variations in circulation across the different cross-sections of the flexible plate ‘F7’ remain limited. The circulation growth curves in the chordwise and spanwise cross-sections are nearly identical, while the diagonal cross-section exhibits slightly lower values. Similarly, for the rigid plate ‘R5’, where the axisymmetric vortex ring is maintained, the circulation growth curves in all three cross-sections almost overlap. In contrast, the extra-flexible plate ‘EF6’ exhibits differences in circulation growth across the cross-sections due to its non-axisymmetry.

Comparing the circulation among different plates, the flexible plate ‘F7’ exhibits circulation values comparable to those of the rigid plate ‘R5’. This suggests that the total vorticity generated by the flexible plate ‘F7’ is similar to that of the rigid plate ‘R5’, which may be attributed to their similar projected shape and area, as shown in figure 7(a,b). Additionally, the flexible plate ‘F7’ exhibits a longer linear growth phase in circulation, extending up to approximately $t^* = 3.8$ , whereas for the rigid plate ‘R5’, the linear growth phase ends at approximately $t^* = 3$ . This prolonged growth phase may be related to the contribution of vortex breakdown. In contrast, the extra-flexible plate ‘EF6’ exhibits significantly lower circulation values than the other two cases. Due to its pronounced bending (figure 7 c), the projected area is nearly halved, leading to a weakened vorticity supply from the plate tip.

Figure 14.

Comparison of the non-dimensional circulation ${\varGamma _v}^*$ over non-dimensional time $t^*$ in the chordwise (red line), diagonal (green line) and spanwise (blue line) cross-sections for plates with three different non-dimensional bending stiffness: (a) ‘R-5’, (b) ‘F-7’ and (c) ‘EF-6’.

4.2.2. Vorticity distribution

The vorticity distribution can be characterized by

(4.3) \begin{equation} r^*_v = \frac {1}{U_cD^2} \int _0^D \int _{-D}^{z_{{\textit{tip}}}^{\prime}} r'\omega \, \mathrm{d}z' \, \mathrm{d}r^{\prime}, \end{equation}

and

(4.4) \begin{equation} z^*_v = \frac {1}{U_cD^2} \int _0^D \int _{-D}^{z_{{\textit{tip}}}^{\prime}} z'\omega \, \mathrm{d}z' \, \mathrm{d}r' - z_{\textit{plate}}^{\prime}/D. \end{equation}

Here, $r^*_v$ is defined as the radial vorticity distribution index, and $z^*_v$ is defined as the axial vorticity distribution index. The term $z_{{plate}}'$ represents the axial position of the plate centre in cylindrical coordinates, corresponding to the $x$ -coordinate in figure 1, and the subtraction accounts for the relative position of the axial vorticity distribution with respect to the plate. The integration domains are identical to those defined in (4.2). While standardly used to locate single vortex cores (Wu, Ma & Zhou Reference Wu, Ma and Zhou2007; Xiang et al. Reference Xiang, Li, Qin and Liu2021), for the complex multicore wakes of flexible/extra-flexible plates (§ 4.1), these indices serve to characterize the global vorticity distribution, thereby quantifying the ring’s 3-D deformation. The computed results are presented in figures 15 and 16.

Figure 15.

Comparison of the radial vorticity distribution index $r^*_v$ over non-dimensional time $t^*$ in the chordwise (red line), diagonal (green line) and spanwise (blue line) cross-sections for plates with three different non-dimensional bending stiffness: (a) ‘R-5’, (b) ‘F-7’ and (c) ‘EF-6’. To verify that the observed radial redistribution is physical, the signal-to-noise ratio over five runs was analysed. For the flexible plate (‘F-7’), the run-to-run variability ( $\sigma _{r_v^*} \approx 0.01$ ) is an order of magnitude smaller than the physical variation during load-shifting ( $\Delta r_v^* \approx 0.12$ ), confirming statistical robustness. Similarly, for the extra-flexible plate (‘EF-6’), despite slightly higher variability due to flow complexity ( $\sigma _{r_v^*} \approx 0.02$ ), the standard deviation remains sufficiently low to clearly distinguish the mean radial trend ( $\Delta r_v^* \approx 0.26$ ) from measurement uncertainty.

Figure 16.

Comparison of the axial vorticity distribution index $z^*_v$ over non-dimensional time $t^*$ in the chordwise (red line), diagonal (green line) and spanwise (blue line) cross-sections for plates with three different non-dimensional bending stiffness: (a) ‘R-5’, (b) ‘F-7’ and (c) ‘EF-6’.

The wake non-axisymmetry of ‘F7’ is quantitatively reflected in the evolution of $r^*_v$ and $z^*_v$ across different cross-sections. Unlike ‘R-5’, where these indices exhibit nearly identical evolution across all three cross-sections (figures 15 a and 16 a), the flexible plate shows distinct variations among them. As shown in figure 15(b), in the chordwise cross-section, $r^*_v$ remains nearly constant, whereas in the diagonal and spanwise cross-sections, it first decreases and then increases. The maximum difference $\Delta r^*_v$ occurs between the chordwise and spanwise cross-sections, reaching approximately 0.12 around $t^* = 3$ . A similar trend is observed in the axial vorticity index (figure 16 b), where the largest discrepancy $\Delta z^*_v$ appears around $t^* = 1.5$ between the chordwise and spanwise cross-sections. These variations highlight the non-axisymmetry in the wake of the flexible plate.

Compared with the flexible plate, the extra-flexible plate ‘EF6’ exhibits even stronger non-axisymmetry, with $\Delta r^*_v$ exceeding 0.15 and $\Delta z^*_v$ exceeding 0.32 across different cross-sections. As shown in figure 15, in the chordwise cross-section, the radial vorticity index $r^*_v$ rapidly decreases to approximately $r^*_v = 0.26$ after the plate’s initial acceleration phase, followed by a secondary increase due to outward vorticity diffusion (as shown in the Supplementary materials), reaching $r^*_v \approx 0.5$ at $t^* = 7$ . In contrast, in the diagonal and spanwise cross-sections, $r^*_v$ decreases after acceleration and remains around 0.3. For the axial vorticity distribution (figure 16), the maximum deviation across chordwise and spanwise cross-section reaches 0.32 at approximately $t^* = 2$ , significantly exceeding the values observed for the flexible plate and indicating stronger wake non-axisymmetry.

Figure 17.

Schematic of the evolution of the jet elliptic vortex ring (a) and the vortex ring behind the flexible plate (b). The schematic in panel (b) is a data-driven visualization of the vortex ring’s 3-D evolution. Its shape and radial contraction are quantitatively supported by the index in figure 15, while its axial bending and lag are supported by the index in figure 16.

Beyond non-axisymmetry, the initial decrease and subsequent increase in the radial vorticity index for the flexible plates warrant further attention. This flow phenomenon is particularly relevant for the load-shifting force phenomenon of the flexible plate, given its circulation is comparable to that of the rigid plate (figure 14). The redistribution of vorticity in the radial direction plays a crucial role in determining the force contribution of the wake, as further discussed in § 5.3.

Additionally, the absolute slope of the axial vorticity index curve is significantly larger for the flexible plate than for the rigid plate (figure 16), indicating enhanced downstream vorticity transport. This may be attributed to vortex breakdown, which facilitates vorticity propagation downstream, unlike the rigid plate, where a single coherent vortex structure persists. For the extra-flexible plate, vortex shedding resembles aerofoil wake dynamics, resulting in even stronger downstream vorticity transport.

4.3. Vortex ring non-axisymmetric evolution and vortex breakdown

Based on the previous discussion, driven by the reconfiguration of the plate, the evolution of the wake vortices behind the flexible plate can be divided into three stages.

The first stage involves the growth and evolution of a single non-axisymmetric vortex ring, as illustrated in figure 17(b). In this phase, the vorticity distribution index can serve as an indicator of the vortex core position. Morphologically, this vortex ring exhibits distinct characteristics. Radially, the vortex ring radius in the chordwise cross-section exceeds that in the spanwise cross-section, forming an elliptical shape when projected onto the $yz$ -plane, with its major axis in the chordwise cross-section and minor axis in the spanwise cross-section, as indicated in figure 15(b). Axially, the vortex core in the chordwise cross-section lags behind that in the spanwise cross-section, leading to a 3-D deformation that mirrors the flexible plate’s shape, as quantified by figure 16(b). As $t^*$ increases, the major axis radius remains relatively constant, whereas the minor axis radius gradually decreases, resulting in a continuous increase in the vortex ring’s aspect ratio. Additionally, the axial displacement $\Delta z^*_r$ between the vortex cores in the spanwise and chordwise cross-sections grows, indicating a progressive enhancement of the vortex ring’s streamwise curvature.

The second stage is the transition phase. During this period, the single vortex ring structure begins to break down (§ 4.1). In the chordwise cross-section, a second shedding vortex emerges and propagates downstream. Meanwhile, in the spanwise cross-section, the vortex cores approach each other, leading to the annihilation of opposite-signed vorticity, which amplifies disturbances and eventually triggers vortex breakdown.

The third stage is the vortex breakdown phase. At this point, the vortex transitions from a clustered distribution to a dispersed distribution (§ 4.1). This dispersed distribution enhances fluid momentum and energy exchange while also promoting vorticity diffusion. Consequently, in the diagonal and spanwise cross-sections, the vorticity, which was previously concentrated near the symmetry axis due to vortex induction, spreads outward into low-vorticity regions. This redistribution leads to an increase in the radial vorticity index $r^*_v$ for $t^*\gt 3$ , by which time vortex breakdown has already occurred. Note that in this study, ‘vortex breakdown’ denotes the rapid disintegration of a coherent vortex ring into a dispersed turbulent state. This usage emphasizes the abrupt loss of vortex coherence, distinct from the classical bubble-type or spiral breakdown observed in strongly swirling flows (Leibovich Reference Leibovich1978; Lucca-Negro & O’Doherty Reference Lucca-Negro and O’Doherty2001).

Having outlined the temporal evolution, the following analysis deconstructs the two most critical features of this process: the non-axisymmetric evolution of the vortex ring and its subsequent breakdown. The former is analysed through the principle of vortex self-induction (§ 4.3.1), whereas the latter is characterized using objective, statistical metrics of flow instability (§ 4.3.2).

4.3.1. Vortex ring non-axisymmetric evolution

Given that the projection area of the flexible plate changes slightly (figure 7 b), where the dimensionless projected area remains nearly 1, the observed non-axisymmetric vortex ring growth is thus primarily attributed to the 3-D reconfiguration of the plate rather than variations in its projected area. As the plate deforms into a 3-D shape, the shed vortex ring inherits this curvature. This process can be better understood using the Biot–Savart law. According to the work of Wu (Reference Wu2021), the self-induced velocity of a vortex filament in the binormal direction $\boldsymbol{b}$ is given by

(4.5) \begin{equation} \boldsymbol{v}_b = \frac {{\varGamma _{\!f}}\kappa _{\!f}}{4\pi } \boldsymbol{b} \ln \frac {L_{\!f}}{\epsilon _{\!f}}, \end{equation}

where ${\varGamma _{\!f}}$ , $\kappa _{\!f}$ , $L_{\!f}$ and $\epsilon _{\!f}$ represent the circulation, curvature, length and vortex core radius of the filament segment, respectively. For a rigid plate, the vortex properties are uniform along the azimuthal direction. Consequently, during its evolution, the vortex ring retains a circular and axisymmetric shape as shown in the Supplementary materials.

In contrast, for flexible plates, plate bending induces 3-D features in the vortex ring. As illustrated in figure 17(b), the vortex ring bends downstream following the plate’s curvature. According to (4.5), the binormal direction $\boldsymbol{b}$ in the chordwise cross-section points away from the vortex ring symmetry axis, whereas in the spanwise cross-section, it points towards the symmetry axis. Consequently, under the self-induced effects of curved vortex filaments, the chordwise-axis segments expand outward while the spanwise-axis segments contract inward. Moreover, throughout the evolution, the binormal direction $\boldsymbol{b}$ in both the chordwise and spanwise cross-sections remains relatively unchanged, causing a sustained outward motion of the vortex filaments in the chordwise cross-section and an inward contraction in the spanwise cross-section.

A comparison with non-axisymmetric jet vortex rings provides further insight into this process. Unlike plate-induced vortex rings, non-axisymmetric jet vortex rings exhibit the well-known axis-switching phenomenon (Cheng, Lou & Lim Reference Cheng, Lou and Lim2016; Fernando & Rival Reference Fernando and Rival2016a ), where the major and minor axes periodically exchange, as shown in figure 17(a). The binormal direction $\boldsymbol{b}$ continuously alternates its orientation relative to the symmetry axis, causing different regions of the filament to expand/contract and accelerate/decelerate in succession. The vortex filament segments with the highest curvature near the major axis propagate faster than the rest of the ring. This forward propagation reduces their curvature while increasing that of the slower-moving regions, eventually leading to a role reversal between the major and minor axes. This process repeats cyclically. In contrast, for flexible plates, axis-switching does not occur – the vortex ring radius in the spanwise plane remains consistently smaller than in the chordwise plane.

4.3.2. Vortex breakdown

To understand the mechanism behind the vortex breakdown observed in the flexible plate’s wake, we begin from the premise that the breakdown is triggered by the growth of a 3-D fluid instability (Saffman Reference Saffman1995; Wu et al. Reference Wu, Ma and Zhou2007). A primary mechanism for the amplification of such instabilities in a vortex is the intense, localized stretching of vortex lines (Adrian, Meinhart & Tomkins Reference Adrian, Meinhart and Tomkins2000; Kundu et al. Reference Kundu, Cohen, Dowling and Capecelatro2024). For an incompressible flow, the intensity of out-of-plane vortex-line stretching can be directly measured from 2-D planar PIV data via the in-plane divergence (Kaiser et al. Reference Kaiser, Kriegseis and Rival2020). We therefore utilize the normalized cylindrical divergence, $\zeta$ , to evaluate the local intensity of this stretching relative to the in-plane fluid deformation (shear and strain) (Reijtenbagh et al. Reference Reijtenbagh, Westerweel and Van de Water2023b ). A detailed derivation and definition of $\zeta$ is provided in Appendix B1.

The instantaneous fields of normalized divergence $\zeta$ and velocity vectors are compared in the chordwise cross-section ( $\theta = 0^\circ$ ) in figure 18. For the rigid plate ‘R-5’ (figure 18 a–c), the $\zeta$ field remains weak and the corresponding velocity vectors illustrate a smooth, stable flow pattern, confirming the wake’s axisymmetry. In contrast, the flexible plate ‘F-7’ (figure 18 d–f) exhibits a dramatic evolution. At $t^*=4$ , strong, spatially organized regions of positive and negative $\zeta$ emerge, coinciding with the onset of vortex breakdown. The velocity vectors in these regions reveal local sources (diverging vectors), which are unambiguous kinematic signatures of the out-of-plane fluid motion associated with intense vortex stretching. For the extra-flexible plate ‘EF-6’ (figure 18 g–i), the wake shows 3-D shear-layer characteristics from the outset; its $\zeta$ field is strong early on but lacks the organized structures that signify the single catastrophic breakdown event seen in the flexible case.

Although the wake is non-axisymmetric, these qualitative characteristics are consistent across the different measurement planes. The flow fields in the diagonal ( $\theta =45^\circ$ , figure 19) and spanwise ( $\theta =90^\circ$ , figure 20) cross-sections likewise show that the rigid plate remains stable, the flexible plate undergoes a distinct transition to a 3-D breakdown and the extra-flexible plate is inherently 3-D throughout its evolution.

Figure 18.

Instantaneous fields of the normalized divergence $\zeta$ (colour contours) and velocity vectors in the chordwise ( $\theta = 0^\circ$ ) cross-section for plates with three different non-dimensional bending stiffness: (a–c) for ‘R-5’, (d–f) for ‘F-7’ and (g–i) for ‘EF-6’. The velocity vectors represent the in-plane components ( $u_{z^{\prime }}^{\prime }, u_{r'}$ ) in the moving frame of reference. Here $u_{z^{\prime }}^{\prime }=u_{z^{\prime }}-U(t)$ is the axial velocity component, where $u_{z^{\prime }}$ is the instantaneous axial flow velocity in the earth frame and $U(t)$ is the instantaneous velocity of the plate. The area enclosed by the green line denotes the region where $\psi _0 \lt 0$ within the mask ${\mathcal{H}}_{{\textit{mask}}}$ (Appendix B.2).

Figure 19.

Instantaneous fields of the normalized divergence $\zeta$ (colour contours) and velocity vectors in the diagonal ( $\theta = 45^\circ$ ) cross-section for plates with three different non-dimensional bending stiffness: (a–c) for ‘R-5’, (d–f) for ‘F-7’ and (g–i) for ‘EF-6’. Annotations are consistent with those in figure 18.

Figure 20.

Instantaneous fields of the normalized divergence $\zeta$ (colour contours) and velocity vectors in the spanwise ( $\theta = 90^\circ$ ) cross-section for plates with three different non-dimensional bending stiffness: (a–c) for ‘R-5’, (d–f) for ‘F-7’ and (g–i) for ‘EF-6’. Annotations are consistent with those in figure 18.

Figure 21.

Comparison of the r.m.s. of the normalized divergence $\zeta _{{\textit{rms}}}$ over non-dimensional time $t^*$ in the chordwise (red line), diagonal (green line) and spanwise (blue line) cross-sections for plates with three different non-dimensional bending stiffness: (a) ‘R-5’, (b) ‘F-7’ and (c) ‘EF-6’.

To statistically quantify the overall evolution of these 3-D effects, the root-mean-square (r.m.s.) of the $\zeta$ field, $\zeta _{{\textit{rms}}}$ , is calculated over the physically relevant regions of the wake (see Appendix B2), as shown in figure 21. The $\zeta _{{\textit{rms}}}$ for the rigid plate remains at a low, baseline level, indicating persistent flow stability. A slight increase is observed around $t^* \approx 5$ , which may be attributed to vortex ring transition to turbulence in the later stage of the plate motion (Li et al. Reference Li, Xiang, Qin, Liu and Wang2022; Reijtenbagh et al. Reference Reijtenbagh, Westerweel and Van de Water2023b ). In contrast, the flexible plate’s $\zeta _{{\textit{rms}}}$ exhibits a distinct change in slope and a rapid increase starting around $t^* \approx 3$ , providing a clear quantitative signature for the transition of the wake from a quasi-2-D state to a fully 3-D, fragmented state. The $\zeta _{{\textit{rms}}}$ for the extra-flexible plate is moderately elevated from the beginning and grows continuously, consistent with its non-transient, 3-D wake. These results lead to the conclusion that the plate’s passive reconfiguration is the root cause of the vortex breakdown. For the flexible plate, plate reconfiguration induces a non-axisymmetric vortex ring (§ 4.3.1) that, under its own self-induced velocity, develops localized regions of intense vortex stretching (high $|\zeta |$ ), ultimately triggering the catastrophic vortex breakdown. The rigid plate avoids this instability due to its axisymmetry, while the extra-flexible plate’s wake is inherently 3-D and does not undergo a distinct breakdown of a coherent structure.

5.1. Force estimation based on the vorticity moment theorem

As discussed in § 1, while the classical quasisteady approach, which decomposes the force into quasisteady, added-mass and history components, is powerful, its history force term is challenging to interpret physically for the complex, large-scale separated flows (Fernando & Rival Reference Fernando and Rival2016a , Reference Fernando and Rivalb ; Li et al. Reference Li, Xiang, Qin, Liu and Wang2022; Liu & Sun Reference Liu and Sun2024). Therefore, to achieve this study’s primary objective of directly linking force generation to the observed wake dynamics, the impulse-based framework is adopted. Rooted in the vorticity moment theorem (Wu Reference Wu1981), this approach is ideal as it directly connects the hydrodynamic force to the evolution of the vorticity field (Kim & Gharib Reference Kim and Gharib2011a , Reference Kim and Gharibb ; Kaiser et al. Reference Kaiser, Kriegseis and Rival2020; Li et al. Reference Li, Xiang, Qin, Liu and Wang2022).

According to the vorticity moment theorem (Wu Reference Wu1981), for a 3-D incompressible and viscous flow in an infinite fluid field, which is at rest at infinity, the force on the body can be decomposed into two parts,

(5.1) \begin{equation} \boldsymbol{F} = \boldsymbol{F}_I + \boldsymbol{F}_b, \end{equation}

where $\boldsymbol{F}_I$ represents the impulsive force and $\boldsymbol{F}_b$ represents the buoyancy force.

The second term $\boldsymbol{F}_b$ in (5.1) is first discussed and defined as

(5.2) \begin{equation} \boldsymbol{F}_b = \rho _{\!f} \frac {{\rm d}}{\mathrm{d}t} \int _{V_b} \boldsymbol{u} \, \mathrm{d}V, \end{equation}

where $\boldsymbol{u}$ is the velocity of the object, and $V_b$ is the volume occupied by the body. According to the kinematic set-up, the plate translates in one direction, and this study focuses on the force in the $x$ -direction. Thus, $\boldsymbol{F}_b$ can be rewritten as $F_b$ , representing the force in the $x$ -direction only. As reference in figure 3, the velocity at the tip and the centre of the plate is nearly identical, suggesting that the velocity at all points on the plate can be considered uniform. And although the shape of the plate changes, its volume remains constant, so $V_b = Sh$ , where $S$ is the area of the plate and $h$ is its thickness. The force $\boldsymbol{F}_b$ can then be expressed as

(5.3) \begin{equation} F_b = \rho _{\!f} \frac {\mathrm{d}U(t)}{\mathrm{d}t} V_b = \rho _{\!f} a(t)Sh, \end{equation}

and the corresponding force coefficient is

(5.4) \begin{equation} C_b = \frac {F_b}{\dfrac {1}{2} \rho U_c^2 S}. \end{equation}

It is noteworthy that $C_b$ only exists during the acceleration phase. In the steady translation phase, where there is no acceleration, $F_b$ becomes zero. For the acceleration phase, substituting further, the final expression can be derived as

(5.5) \begin{equation} C_b = 2a^* \frac {h}{D}. \end{equation}

In this study, $a^* = 0.25$ is a constant, and the maximum value of $h$ is $3.00$ mm. Consequently, $C_b \leqslant 0.015$ .

The first term $\boldsymbol{F}_I$ in (5.1) is calculated as

(5.6) \begin{equation} \boldsymbol{F}_I = -\frac {\rho _{\!f}}{2} \frac {\mathrm{d} \boldsymbol{I}}{\mathrm{d}t}, \end{equation}

where,

(5.7) \begin{equation} \boldsymbol{I} = \int _{V_{\infty }} \boldsymbol{x} \times \boldsymbol{\omega } \, \mathrm{d}V, \end{equation}

is the first moment of the combined fluid-body vorticity field. Here, $\boldsymbol{x}$ is the position vector, $\boldsymbol{\omega }$ is the vorticity and $V_\infty = V_{\!f} + V_b$ is the volume occupied by the fluid and the body. For the translating plate without rotation in this study, the integration of $V_b$ vanishes (Graham Reference Graham2019), leading to

(5.8) \begin{equation} \boldsymbol{I} = \int _{V_{\!f}} \boldsymbol{x} \times \boldsymbol{\omega } \, \mathrm{d}V. \end{equation}

As a widely accepted theory, any object moving in the real viscous fluid can be represented by a vortex sheet located on its surface, which models the viscous boundary layer (Saffman Reference Saffman1995). Referring to Corkery (Reference Corkery2019) and Gehlert (Reference Gehlert2021), the complex viscous flow can be represented as the vortex sheet in the potential flow by superposing two independent components. One contribution arises from the inviscid flow around the plate during translation at an angle of incidence of $90^\circ$ , while the other contribution comes from the shed vorticity behind the plate. The former generates an added mass vortex sheet, while the latter induces a circulatory vortex sheet. Thus, the decomposition of (5.8) aligns with the physical interpretation of the vortex sheet,

(5.9) \begin{equation} I = I_a + I_v, \end{equation}

where, as this study focuses on the force in the $x$ -direction, $\boldsymbol{I}$ can be rewritten as $I$ , representing the force in the $x$ -direction only; $I_a$ corresponds to the impulse due to the vortex sheet associated with the plate translation, and $I_v$ corresponds to the impulse due to shed vorticity.

According to the results of Corkery, Babinsky & Graham (Reference Corkery, Babinsky and Graham2019), $I_a$ has the same form in both potential and viscous fluids. Based on this consistency, the added mass model in potential flow can be directly utilized,

(5.10) \begin{equation} C_a \text{ in viscous flow} = C_a \text{ in potential flow}. \end{equation}

Referring to Brennen (Reference Brennen1982), this paper approximates the added mass force of the reconfigurable plate using the added mass force of an elliptical plate. The specific calculation of $C_a$ is detailed in Appendix C.

The calculation of $I_v$ requires consideration of the shed vorticity in the fluid. In cylindrical coordinates,

(5.11) \begin{equation} I_v = \frac {\rho }{2} \int _{V_{\!f}} r' \omega _\theta \, \mathrm{d}V, \end{equation}

where $\omega _\theta$ is the vorticity in the azimuthal direction, and $V_{\!f}$ represents the wake flow where vorticity is shed. Expanding the integration in cylindrical coordinates,

(5.12) \begin{equation} I_v = \frac {\rho }{2} \frac {{\rm d}}{\mathrm{d}t} \int _{z'_{-\infty }}^{z'_{+\infty }} \int _{0}^{2\pi } \int _{0}^{r'_\infty } r'^2 \omega _\theta \, \mathrm{d}r' \, \mathrm{d}\theta \, \mathrm{d}z', \end{equation}

where $z'$ represents the axial direction in cylindrical coordinates, corresponding to the $x$ -axis in the Cartesian coordinate system (see figure 1). For axisymmetric vortex rings, where $\omega _\theta$ is constant along the azimuth, integrating over $\theta$ yields

(5.13) \begin{equation} I_v = \rho \pi \frac {{\rm d}}{\mathrm{d}t} \int _{z'_{-\infty }}^{z'_{+\infty }} \int _{0}^{r'_\infty } r'^2 \omega _\theta \, \mathrm{d}r' \, \mathrm{d}z'. \end{equation}

In this experiment, due to the non-axisymmetric evolution of vortex rings, the assumption that the circulation $\omega _\theta$ remains constant along the circumferential direction does not hold. In practice, the impulse can be calculated by averaging the data across multiple cross-sections passing through the centre, as follows:

(5.14) \begin{equation} I_v \approx \frac {1}{N} \sum _{i=1}^{N} \rho \pi \frac {{\rm d}}{\mathrm{d}t} \int _{z'_{-\infty }}^{z'_{+\infty }} \int _{0}^{r'_\infty } r'^2 \omega _{\theta ,i} \, \mathrm{d}r' \, \mathrm{d}z', \end{equation}

where $i$ represents the $i$ th cross-section, $N$ is the total number of cross-sections, and a higher number of cross-sections improves the accuracy of the impulse calculation. In this study, the global impulse is estimated by averaging data from three representative cross-sections (chordwise, diagonal and spanwise). For a sensitivity analysis regarding the selection of measurement planes, please refer to Appendix A4. The integration domain in each cross-section is identical to that defined in (4.2). Based on this, the force coefficient associated with the shed vorticity can be calculated as

(5.15) \begin{equation} C_v = \frac {1}{q_{\infty } S} \frac {\mathrm{d}I_v}{\mathrm{d}t}. \end{equation}

In summary, the total force coefficient can be expressed as

(5.16) \begin{equation} C_{H,{\textit{impulse}}} = C_v + C_a + C_b. \end{equation}

As shown in figure 22, the impulse-based calculation ( $C_{H,{\textit{impulse}}}$ ) exhibits strong quantitative agreement with the transducer measurement ( $C_{H,{transducer}}$ ). For the rigid and flexible plates, the Pearson correlation coefficient (PCC) exceeds 0.94, confirming that the $C_{H,{\textit{impulse}}}$ accurately captures the temporal evolution trends. Crucially, the relative r.m.s. (RRMSE) within the peak force remains below 6 % for these cases. This high fidelity ensures that the load-shifting characteristics – specifically the reduced peak at $t^*=2$ and the elevated valley at $t^*=5.5$ for the flexible plate – are physically resolved. Notably, even for the extra-flexible plate where 3-D asymmetry is intensified, the peak load is captured with a relative error of less than 8 %. This statistical robustness validates the vorticity moment theorem as a reliable framework for force analysis.

Figure 23 further illustrates the force decomposition results based on vorticity moment theorem. As shown in the figure, $C_v$ dominates the overall contribution, regardless of whether the plate is rigid, flexible or extra-flexible. Given the strong correlation between $C_v$ and the growth and evolution of the vortex ring, these results indicate that, under the parameter set-up in this study, the vorticity evolution serves as the key mechanism driving the force response.

Figure 22.

Comparison between $C_{H,{transducer}}$ measured by the force transducer (red line) and $C_{H,{\textit{impulse}}}$ based on the vorticity moment theorem (green line) for three representative cases: (a) ‘R-5’, (b) ‘F-7’ and (c) ‘EF-6’.

Figure 23.

Force decomposition based on the vorticity moment theorem, showing $C_{v,i}$ (dash–dotted line), $C_{a}$ (dotted line) and $C_{b}$ (dashed line) for three different non-dimensional bending stiffness values: (a) ‘R-5’, (b) ‘F-7’ and (c) ‘EF-6’. The force decomposition corresponds to $C_{H,{impulse}}$ in figure 22.

5.2. Low-order force model for impulse force from shed vorticity

To analyse the dominant force from shed vorticity, we employ a low-order model that treats the wake as a thin, closed vortex loop (Wu et al. Reference Wu, Ma and Zhou2007; Kim & Gharib Reference Kim and Gharib2011b ; Li et al. Reference Li, Xiang, Qin, Liu and Wang2022). This allows the impulse of the shed vorticity ( $I_v$ ) to be rewritten as

(5.17) \begin{equation} I_v \approx \rho {\varGamma }_v S_v ,\end{equation}

where the circulation ${\varGamma }_v$ and the vortex loop area $S_v$ are calculated as

(5.18) \begin{equation} {\varGamma }_v = \frac {1}{N} \sum _{i=1}^{N} {\varGamma }_i, \end{equation}

and

(5.19) \begin{equation} S_v = \frac {1}{N} \pi \sum _{i=1}^{N} {r_i}^2, \end{equation}

where, ${\varGamma }_i$ and $r_i$ are the circulation and radial vorticity distribution in the $i$ th cross-section, respectively. Similar to the calculations in (5.14), increasing the number of selected cross-sections improves the accuracy of the results. In this study, flow field data from chordwise, diagonal and spanwise cross-sections, are used to compute the results. The normalized values of ${\varGamma }_i$ and $r_i$ for these three cross-sections are shown in figures 14 and 15. By differentiating (5.17), the force coefficient associated with the time derivative of circulation is expressed as

(5.20) \begin{equation} C_{v,{\varGamma }_v} = \frac {1}{\dfrac {1}{2} U_c^2 S} S_v \frac {\mathrm{d}{\varGamma }_v}{\mathrm{d}t}, \end{equation}

and the force coefficient associated with the time derivative of vortex loop area is

(5.21) \begin{equation} C_{v,S_v} = \frac {1}{\dfrac {1}{2} U_c^2 S} {\varGamma }_v \frac {\mathrm{d}S_v}{\mathrm{d}t}. \end{equation}

Based on these, the low-order force model for impulse force from shed vorticity can be expressed as

(5.22) \begin{equation} C_{v, m} = C_{v,{\varGamma }_v} + C_{v,S_v}, \end{equation}

where $C_{v, m}$ represents the force coefficient obtained from the low-order model. The predicted results of $C_{v, m}$ are presented in figure 24(a–c). Quantitative analysis reveals that for the rigid and flexible plates, $C_{v, m}$ exhibits strong alignment with $C_{v, i}$ , characterized by PCC ${\gt } 0.75$ and RRMSE ${\lt} 20\,\%$ . This statistical agreement confirms that the low-order model effectively captures the key force characteristics in regimes governed by coherent vortex dynamics. More error analysis can be found in Appendix D.

Figure 24(d–f) further shows the decomposition results of $C_{v, m}$ for different ${\textit{EI}}^*$ plates.

For rigid plate (figure 24 d), the contribution of $C_{v,{\varGamma }_v}$ is dominant, while that of $C_{v,S_v}$ is secondary. The contribution from $C_{v,S_v}$ , which is proportional to the rate of change of the vortex area ( $\mathrm{d}S_v/\mathrm{d}t$ ) as defined in (5.21), remains negligible because the radial vorticity distribution stays nearly constant during the motion (figure 15), leading to an area change rate close to zero. In contrast, $C_{v,{\varGamma }_v}$ governs the overall force evolution, producing the characteristic peak around $t^* = 2$ and the subsequent trough near $t^* \approx 5.5$ . This is because $C_{v,{\varGamma }_v}$ is proportional to the circulation growth rate ( $\mathrm{d}{\varGamma }_v/\mathrm{d}t$ ) as shown in (5.20), and the circulation undergoes a clear and significant increase over time (figure 14). Therefore, for the rigid plate, the force variation is almost entirely driven by the evolution of vortex circulation – an observation consistent with previous studies (Wu et al. Reference Wu, Ma and Zhou2007; Kim & Gharib Reference Kim and Gharib2011b ; Li et al. Reference Li, Xiang, Qin, Liu and Wang2022).

For the flexible plate (figure 24 e), the contribution of $C_{v,{\varGamma }_v}$ decreases during the acceleration phase, while $C_{v,S_v}$ exhibits a negative value in this phase and significantly increases after the acceleration ends. These changes lead to the load-shifting phenomenon observed in the force evolution of the flexible plate, which will be further discussed in § 5.3.

The absence of load-shifting phenomenon for extra-flexible plates can be well explained using figure 24(f). First, the contributions of $C_{v,{\varGamma }_v}$ and $C_{v,S_v}$ for the extra-flexible plate are relatively small, leading to a smaller overall force. Second, $C_{v,{\varGamma }_v}$ and $C_{v,S_v}$ complement each other, resulting in a relatively stable force evolution without distinct peaks and troughs.

Figure 24.

Comparison between the impulse-based coefficient $C_{v,i}$ (red solid line) and the model-based coefficient $C_{v,m}$ (green solid line) for three representative cases with varying non-dimensional bending stiffness: (a,d) ‘R-5’, (b,e) ‘F-7’ and (c,f) ‘EF-6’. The values of $C_{v,i}$ shown here correspond to those presented in figure 23. Panels (d)–(f) further decompose $C_{v,m}$ into its two components: $C_{v,{\varGamma }_v}$ (green dotted line) and $C_{v,S_v}$ (blue dash–dotted line). Quantitative metrics highlight the regime transition for panels (a)–(c): the rigid case shows an excellent fit (PCC $= 0.98$ , RRMSE ${\lt} 10\,\%$ ); the flexible case maintains reasonable agreement (PCC $= 0.75$ , RRMSE ${\lt} 20\,\%$ ); whereas the extra-flexible case exhibits deviation (PCC $= 0.68$ , RRMSE ${\lt} 36\,\%$ ).

5.3. Mechanism for the load-shifting phenomenon

The low-order force model accurately predicts the impulse force generated by shed vorticity. By decomposing the impulse force into its constituent components ((5.20) and (5.21)) and synthesizing these with the evolution of ${\varGamma }_v^*$ (figure 14) and $r_v^*$ (figure 15), the model provides a mechanistic explanation for the load-shifting phenomenon.

The load-shifting phenomenon is characterized by two distinct features when comparing the flexible plate to the rigid one: a reduced force peak at the end of acceleration ( $t^* = 2$ ) and an elevated force valley in the postacceleration phase ( $t^* \approx 5.5$ ) (figures 7 and 8). The analysis begins with the full decomposition of the total hydrodynamic force into components from shed vorticity ( $C_v$ ), added mass ( $C_a$ ) and buoyancy ( $C_b$ ) (5.16). As shown in figure 23, the force from shed vorticity, $C_v$ , is the overwhelmingly dominant component. Therefore, to understand the mechanism, the analysis must focus on the composition of $C_v$ . To this end, the low-order model is employed (5.22) to further decompose $C_v$ into two terms: $C_{v,{\varGamma }_v}$ (5.20) and $C_{v,S_v}$ (5.21).

First, the mechanism behind the reduced peak force of the flexible plate is examined. At the force peak ( $t^* = 2$ ), figure 24 shows that both $C_{v,{\varGamma }_v}$ and $C_{v,S_v}$ are smaller than those of the rigid plate, with $C_{v,S_v}$ even exhibiting a negative value. According to (5.20), the force component $C_{v,{\varGamma }_v}$ is governed by the product of the vortex loop area $S_v$ and the circulation growth rate $\mathrm{d}{\varGamma }_v^*/\mathrm{d}t$ . Although the flexible plate has a slightly higher circulation growth rate (figure 14), its $C_{v,{\varGamma }_v}$ is smaller than that of the rigid plate (figure 24 e versus figure 24 d), due to a significantly smaller vortex loop area $S_v$ , which scales with $(r_v^*)^2$ (figure 15). Similarly, the negative value of $C_{v,S_v}$ can be explained by the definition in (5.21). With the flexible plate having larger circulation (figure 14), the observed negative $C_{v,S_v}$ must result from a decreasing radial vorticity distribution, i.e. $\mathrm{d}r_v^*/\mathrm{d}t \lt 0$ (figure 15). Together, these results demonstrate that the smaller radial vorticity distribution $r_v^*$ and its negative temporal change dominate the reduction in peak force. Therefore, the radial vorticity distribution is the key mechanism responsible for the reduced peak force.

Next, the mechanism behind the elevated force valley in the flexible plate is analysed. As shown in figure 24, this postpeak phase ( $t^* \approx 5.5$ ) is dominated by the $C_{v,S_v}$ component, which is large and positive for the flexible plate but near-zero for the rigid plate; in contrast, the $C_{v,\varGamma _v}$ component is small for both cases at this time. According to (5.21), as the total circulation, ${\varGamma }_v^*$ , is comparable for both plates (figure 14), the larger $C_{v,S_v}$ in the flexible case must be caused by a greater vortex area growth rate ( $\mathrm{d}S_v/\mathrm{d}t$ ). Given that the vortex area, $S_v$ , scales with $(r_v^*)^2$ , this larger $\mathrm{d}S_v/\mathrm{d}t$ directly reflects an outward movement of the vortex ring ( $\mathrm{d}r_v^*/\mathrm{d}t \gt 0$ ), a trend confirmed by the vorticity distribution data in figure 15. Thus, the radial redistribution of vorticity is also the key mechanism responsible for elevating the force in the postpeak valley.

In summary, the load-shifting phenomenon – manifested as a reduced peak force at $t^* = 2$ and an elevated force valley at $t^* = 5.5$ in the flexible plate – is primarily driven by the radial vorticity distribution ( $r_v^*$ ). While the circulation ${\varGamma }_v^*$ directly contributes to the overall force magnitude and dominates the formation of the peak at $t^* = 2$ , it does not explain the difference between the flexible and rigid plates. Furthermore, based on the consistency of the vorticity dispersion length across different plates shown in Appendix D, we also rule out the potential impact of vorticity dispersion on load-shifting. Combining the discussion of § 4.3, this analysis further highlights the critical role of structural reconfiguration in shaping the vortex-force interaction through radial redistribution.

Figure 25 further illustrates the force decomposition results for varying Reynolds numbers at the fixed ${\textit{EI}}^*=1.16\times 10^{-2}$ . Although the overall magnitude of $C_{v,m}$ remains similar, the relative contributions of $C_{v,{\varGamma }_v}$ and $C_{v,S_v}$ vary. As the Reynolds number increases, $C_{v,{\varGamma }_v}$ decreases more significantly around $t^* = 5.5$ , whereas $C_{v,S_v}$ becomes more dominant. As discussed in the Supplementary materials, this trend likely stems from enhanced vortex breakdown at higher Reynolds numbers, which promotes stronger radial vorticity redistribution and further amplifies the load-shifting behaviour in flexible plates.

Figure 25.

Comparison (a–c) between $C_{v,i}$ based on the vorticity moment theorem (red line) and $C_{v,m}$ based on the low-order model (green line), and the decomposition (d–f) of $C_{v,m}$ , $C_{v,{\varGamma }_v}$ (green dotted line) and $C_{v,S_v}$ (blue dash–dotted line) with non-dimensional bending stiffness ${\textit{EI}}^*=1.18\times 10^{-2}$ but three different Reynolds numbers: (a,d) ‘F-3’ , (b,e) ‘F-7’ and (c, f) ‘F-10’.