4.1. Structural response modes

Through numerical simulations over a range of key parameters, including the dimensionless bending stiffness, the Keulegan–Carpenter number and the inclination angle, the response behaviours of a thin flexible plate in oscillatory flow can be classified into four modes: static mode, flapping mode, subharmonic flapping mode and irregular mode. The superpositions of the plate profiles together with their time histories of plate tip deflection angle and the PSD with respect to the deflection are shown in figure 6. The superposed plate profiles are plotted over four consecutive oscillatory flow periods after the plate response behaviours have reached a steady state. The tip deflection angle of the plate is measured by the straight line connecting the plate’s leading and trailing edges with respect to its initial undeformed position. The power spectra are analysed through the same time interval as that used for the deflection time histories.

Figure 6.

Superpositions of plate profiles, tip deflection angle histories and power spectral density (PSD) for (a) static mode ( $K_{\!B} = 4.0$ , ${\textit{KC}} = \pi$ and $\alpha = 5^{\circ }$ ), (b) flapping mode ( $K_{\!B} = 0.5$ , ${\textit{KC}} = \pi$ and $\alpha = 5^{\circ }$ ), (c) subharmonic flapping mode ( $K_{\!B} = 0.2$ , ${\textit{KC}} = \pi$ and $\alpha = 5^{\circ }$ ) and (d) irregular mode ( $K_{\!B} = 0.05$ , ${\textit{KC}} = \pi$ and $\alpha = 5^{\circ }$ ).

In figure 6(a i), the static mode at a high dimensionless bending stiffness is shown, where the plate remains straight and undeformed due to the large elastic resistance against the fluid force. The temporal variation of the tip deflection angle shows a negligible deviation from zero, which is consistent with the plate profile. Multiple frequency peaks are observed for the static mode in figure 6(a iii). When the dimensionless bending stiffness decreases to $K_{\!B} = 0.5$ , the structural response mode switches to the flapping mode, as shown in figure 6(b i), where the plate oscillates around its equilibrium position in a regular and periodic motion with an approximately constant amplitude and frequency (see figure 6 b ii). The plate profiles over each oscillatory flow period are perfectly overlapping, and the response frequency is dominated by the external oscillatory flow frequency. The flapping mode is governed by the driven fluid pressure force and restoring elastic force, which resembles the large-amplitude flapping of an inverted flexible plate in uniform flow (Sader et al. Reference Sader, Cosse, Kim, Fan and Gharib2016a ). In figure 6(c ii), with $K_{\!B}$ declining to $0.2$ , the plate shows a similar flapping motion but with a larger flapping amplitude. However, in a single flapping period, the plate oscillates with a deflected mean position, and the mean position changes its side in two continuous periods. The switching of the mean deflected position is clearly identified from the plate profiles shown in figure 6(c i), which are distinguished by different colours. Consequently, the dominant flapping frequency is half of the oscillating flow frequency, as seen in figure 6(c iii). Therefore, in the current research, this typical structural response is referred to as the subharmonic flapping mode. Once the dimensionless bending stiffness is low enough ( $K_{\!B} = 0.05$ ), the plate flexibility is unable to provide enough resistance and resilience against the fluid force. As a result, the plate shows an irregular mode, in which the plate flapping becomes irregular and loses the periodicity as observed from figure 6(d i,d ii). The flapping frequency spectra exhibit a primary peak at the fluid oscillation frequency, accompanied by multiple subordinate peaks, particularly in the low-frequency range. The plate kinematics and flow distributions are presented in the movies provided as Supplementary material.

Figure 7.

Phase diagrams of inverted plates in oscillatory flow with respect to $K_{\!B}$ and $\alpha$ under different wave orbital excursion: (a) ${\textit{KC}} = 2 \pi$ , (b) ${\textit{KC}} = \pi$ and (c) ${\textit{KC}} = \pi /2$ .

To investigate the variation of structural response modes of inclined thin flexible plates in oscillatory flows, we conduct a parametric sweep over the Keulegan–Carpenter number ( ${\textit{KC}}$ ), the plate flexibility ( $K_{\!B}$ ) and the inclination angle relative to the flow ( $\alpha$ ). The resulting phase diagrams in the $K_{\!B}$ – $\alpha$ plane for three representative cases, ${\textit{KC}} = \pi /2$ , $\pi$ and $2\pi$ , are shown in figure 7. For all three values of ${\textit{KC}}$ , the static mode is observed in the region of relatively large dimensionless bending stiffness $K_{\!B}$ . As $K_{\!B}$ decreases at a fixed inclination angle, a transition from the static mode to a flapping mode occurs. Furthermore, the transition boundary shifts towards higher $K_{\!B}$ with increasing $\alpha$ . This trend arises because a larger inclination angle exposes a greater frontal area to the oscillatory flow, leading to stronger pressure forces and an enhanced tendency towards static-divergence instability (Gurugubelli & Jaiman Reference Gurugubelli and Jaiman2015).

With a further decrease in bending stiffness, both subharmonic flapping mode and irregular mode emerge, depending on the parameter combinations. The subharmonic mode is confined to a very narrow range of bending stiffness, as shown in figure 7. It is worth noting that, unlike vertically wall-mounted flexible plates in oscillatory flows (Zhang & Nakamura Reference Zhang and Nakamura2024a ), for which the inclination angle relative to the flow is effectively $90^{\circ }$ , an inclined flexible plate does not exhibit nonlinear modes even at $\alpha = 90^{\circ }$ . This difference arises because the trailing edge of the inclined plate is located in an open domain, allowing the flow to shear past the plate without the confinement and ground effect induced by a nearby wall.

4.2. Vortex dynamics

For a better understanding of the interaction between the inclined flexible plates and external oscillating flow, we plot the instantaneous vorticity contour over one fluid oscillating period with varying bending stiffness for $\alpha = 5^{\circ }$ in figure 8 and $\alpha = 60^{\circ }$ in figure 9. Figure 8(a) shows the vorticity evolution for one flow period at ${\textit{KC}} = \pi$ , $K_{\!B} = 4.0$ and $\alpha = 5^{\circ }$ . During the rightward stroke of oscillating flow, a shear layer of positive vorticity forms on the upper surface of the plate, while a negative shear layer develops on the lower surface. After $t/T = 0.5$ , corresponding to flow reversal, the signs of the upper and lower shear layers switch accordingly. Owing to the small inclination angle, the shear layers remain attached to the plate surfaces throughout the oscillation cycle, without flow separation or vortex shedding. This behaviour is consistent with previous numerical studies (Gurugubelli & Jaiman Reference Gurugubelli and Jaiman2015).

Figure 8.

Vorticity contour of inclined thin flexible plates in oscillatory flows at ${\textit{KC}} = \pi$ and $\alpha = 5^{\circ }$ with (a) $K_{\!B} = 4.0$ , (b) $K_{\!B} = 0.5$ , (c) $K_{\!B} = 0.2$ and (d) $K_{\!B} = 0.05$ . The subpanels from (i) to (iv) correspond to the different time moments $t/T = 0.0, 0.25, 0.5$ and $0.75$ , respectively. The plate kinematics and flow distributions are presented in the movies provided as Supplementary material.

Figure 9.

Vorticity contour of inclined thin flexible plates in oscillatory flows at ${\textit{KC}} = \pi$ and $\alpha = 60^{\circ }$ with (a) $K_{\!B} = 4.0$ , (b) $K_{\!B} = 0.5$ , (c) $K_{\!B} = 0.2$ and (d) $K_{\!B} = 0.05$ . The subpanels from (i) to (iv) correspond to the different time moments $t/T = 0.0, 0.25, 0.5$ and $0.75$ , respectively. The plate kinematics and flow distributions are presented in the movies provided as Supplementary material.

In figure 8(b) of the plate at $K_{\!B} = 0.5$ , the plate loses the divergent stability and starts to flap under the external fluid loading. Although the general vorticity evolution is similar to the case of $K_{\!B} = 4.0$ (figure 8 a), the intensity of the vorticity attached to the plate surface is strengthened with a thicker shear layer. Unlike the independent development of the upper and lower shear layers at $K_{\!B} = 4.0$ , here a shear layer interaction is spotted. Once the left-hand stroke proceeds to the right-hand stroke, the negative vorticity filament at the leading edge is pushed rightwards and reattached to the lower surface, which merges with the lower shear layer and creates a vorticity bulge, as seen from figure 8(b ii).

When the dimensionless bending stiffness decreases to $K_{\!B} = 0.2$ , as shown in figure 8(c), the response mode of the plate transits to the subharmonic flapping mode, which offers a significantly larger instantaneous angle of attack due to the drastic structure deformation. Since a single structural flapping period in the subharmonic flapping mode spans two fluid oscillation periods, the illustrations shown here present only the upward flapping phase of the plate. It is clearly shown that the generated shearing vorticity layer on the plate surface is shed from the plate’s leading edge. Consequently, a vortex street consisting of pairs of vortices with opposite signs is produced along the direction of the plate’s leading edge. From the plate trailing edge, another pair of vortices of lower vorticity is shed out and dissipates in the bulk flow. For the plate of irregular mode, see figure 8(d), the extreme compliance and softness of the structure mitigate its resistance to the flow and consequent vorticity shearing. As a result, the shedding of a regular vortex street diminishes; instead, only a small vortex is produced at the plate leading edge, seen from figure 8(d iv).

Differently, for the thin flexible plate with a larger inclination angle, placed in the oscillatory flow, for instance, $\alpha = 60^{\circ }$ in figure 9, the vorticity evolution shows a distinctive pattern compared with the low inclination case, considering two factual differences. First, the large inclination angle causes a stronger shearing effect and flow separation. Second, a wider frontal area associated with the larger inclination angle results in a higher fluid pressure force, which drives the plate into deformation to a greater extent. Figure 9(a) shows the vorticity contour around the plate of bending stiffness $K_{\!B} = 4.0$ with ${\textit{KC}} = \pi$ and $\alpha = 60^{\circ }$ . Two pairs of vortex dipoles are generated and shed into the ambient flow, of which one pair is detached from the leading edge while the other comes from the trailing edge. In the right stroke, a clockwise vortex is generated and shed from the leading edge at $t/T = 0$ and $1/4$ , see figure 9(a i,a ii). However, the flow reverses its direction from right to left at $t/T = 1/2$ and thus pushes the shed clockwise vortex close to the plate surface. Later, the negative shear layer resulting from the strong flow shearing combines with the clockwise vortex and sheds into the bulk, forming a vortex dipole with a tight connection. It is also noted that, during vortex dipole shedding at the leading edge, a vorticity filament forms and subsequently develops into a secondary vortex of smaller scale, which then merges with the trailing edge vortex, see figure 9(a iv). For the static mode at large structural flexibility, due to the negligible degrees of freedom to plate movement, the vorticity evolution at the leading and trailing edge exhibits a highly similar processes, with only slight differences in the dipole propagation direction.

In figure 9(b), the plate response mode transitions to the flapping mode due to the reduced bending stiffness. The alignment of the bending motion with the externally driven flow mitigates the shearing effect, thereby delaying the generation of the leading edge vortex, as evidenced by a comparison between figures 9(b ii,b iii) and 9(a ii,a iii). The aligning flapping reduces the relative velocity between the structure and the bulk flow; therefore, as shown in figure 9(b iv), only a vorticity filament is shed from the leading edge instead of a fully developed vortex, which is caused by the reduced flow shearing. The vorticity filament subsequently grows into an anticlockwise vortex, which further spreads into the far field, accompanied by the clockwise vortex from the last stroke. At the trailing edge of the plate at $K_{\!B} = 0.5$ , although the vorticity filaments of both signs detach from the plate due to the sharp edge, no complete vortices are spotted since the filament either merges with the shear layer at the plate surface or dissipates in the background flow. In figure 9(c), for the case of $K_{\!B} = 0.2$ , the vorticity evolution pattern reverses compared with the $K_{\!B} = 0.5$ case, where the shedding of vortex pairs occurs only at the trailing edge. The upper part of the plate suffers a slighter flow shear and separation due to the large deformation under lower flexibility. A similar vorticity contour can be found in figure 9(d) for the irregular mode, even under a chaotic and aperiodic structural response.

4.3. Lift coefficients of rigid plates in oscillatory flow

The interaction between the flexible structure and the background flow, particularly the associated structural response behaviours, is governed by the balance between external fluid loading and internal structural restoring forces arising from flexibility. The fluid force depends strongly on the relative velocity, frontal area and drag and lift coefficients, which are in turn determined by the structural shape and its degree of streamlining. In this section, we derive the variations of the force coefficients for rigid plates with different inclination angles in oscillatory flow.

Figure 10.

(a) Drag coefficients $C_d$ and (b) lift coefficients $C_l$ of inclined rigid plates in oscillatory flows under various Keulegan–Carpenter numbers, ${\textit{KC}}$ , and inclination angles, $\alpha$ .

For rigid structures immersed in oscillatory flow, extensive experimental and theoretical studies have been conducted on cylinders and vertical plates (Keulegan & Carpenter Reference Keulegan and Carpenter1958; Wang Reference Wang1968). The drag coefficients of a rigid structure immersed in oscillatory flow can be expressed as follows:

(4.1) \begin{equation} C_d = \frac {3}{4} \int _{0}^{2\pi } \frac {F_d \textrm {cos} \theta }{\rho _{\!f} U_0^2 l} \textrm {d} \theta , \end{equation}

where $F_d$ is the horizontal component of the hydrodynamic force exerted on the plate and $\theta$ is the instantaneous phase. Figure 10(a) shows the calculated drag coefficients by (4.1) as a function of ${\textit{KC}}$ with varying inclination angles. As the inclination angle increases, the drag coefficient increases in value, as the plate exposes a wider frontal area and receives higher fluid loading. For the wall-mounted immersed structure in the oscillatory flow, the variation of the drag coefficient follows the scaling law as $C_d \sim KC^{-1}Ro^{-1/2}$ (Keulegan & Carpenter Reference Keulegan and Carpenter1958; Zhang & Nakamura Reference Zhang and Nakamura2024a ), where $Ro = \rho _{\!f} l^2 f / \mu$ is the Roshko number (Williamson & Brown Reference Williamson and Brown1998). On the other hand, when the structure undergoes the oscillatory flow effect in an open domain, the variation of the drag coefficient follows the scaling law of $C_d \sim KC^{-1/2}Re^{-1/2}$ (Keulegan & Carpenter Reference Keulegan and Carpenter1958). As shown in figure 10(a), under the fixed viscous effect ${\textit{Re}} = 200$ , the drag coefficient is proportional to the inverse square root of ${\textit{KC}}$ , $C_d \sim KC^{-1/2}$ for each inclination angle, which is consistent with the previous research results.

However, in previous studies (Keulegan & Carpenter Reference Keulegan and Carpenter1958; Wang Reference Wang1968), only expressions for the drag coefficient were derived, as these works focused on cylinders or vertical plates. Owing to their geometric symmetry with respect to the flow direction, such configurations theoretically generate no lift force. When a plate is placed in oscillatory flow at a finite inclination angle $\alpha \ne 0^{\circ }$ and $\alpha \ne 90^{\circ }$ , a lift force arises naturally due to the broken symmetry, with its direction changing in association with flow reversal. To quantitatively describe the effect of inclination on the lift force, we derive the lift coefficient of an inclined rigid plate in oscillatory flow to characterise the lift force variation.

First, we consider the instantaneous lift force acting on a plate with an angle of attack $\alpha$ (i.e. the inclination angle) in a uniform flow with fluid velocity $U$ . The lift force can be expressed as

(4.2) \begin{equation} F_l = \frac {1}{2} \rho _{\!f} l C_l (\alpha ) U(t)|U(t)|. \end{equation}

Once we substitute the velocity profile of the oscillatory flow, $U(t) = U_0 \textrm {cos} (2 \pi ft)$ , into (4.2), we can have the dimensionless instantaneous lift force within oscillatory flow as

(4.3) \begin{equation} \frac {F_l}{\rho _{\!f} U_0^2 l} = \frac {C_l}{2}|\textrm {cos}\theta |\textrm {cos}\theta , \end{equation}

where $\theta$ is the instantaneous phase, $\theta = 2 \pi \textit{ft}$ .

Note that the lift force exhibits half-period antisymmetry, as once the oscillatory flow reverses from the right-hand stroke to the left-hand stroke, the direction of the lift force changes from upwards to downwards. Owing to the flow symmetry, the lift force satisfies the following relation:

(4.4) \begin{equation} F_l(\theta ) = -F_l(\theta +\pi ). \end{equation}

For the half-period antisymmetric function, using Fourier series expansion, we can have

(4.5) \begin{align} \frac {F_l}{\rho _{\!f} U_0^2 l} &= A_1\textrm {sin}\theta + A_3\textrm {sin}3\theta + A_5\textrm {sin}5\theta + \ldots \nonumber\\ &\quad + B_1\textrm {cos}\theta + B_3\textrm {cos}3\theta + B_5\textrm {cos}5\theta + \ldots , \end{align}

where the coefficients in the Fourier series expansion are

(4.6) \begin{equation} A_n = \frac {1}{\pi } \int _{0}^{2 \pi } \frac {F_l \textrm {sin} n\theta }{\rho _{\!f} U_0^2 l} \textrm {d} \theta , \end{equation}

and

(4.7) \begin{equation} B_n = \frac {1}{\pi } \int _{0}^{2 \pi } \frac {F_l \textrm {cos} n\theta }{\rho _{\!f} U_0^2 l} \textrm {d} \theta . \end{equation}

On the other hand, the expression $|\textrm {cos}\theta |\textrm {cos}\theta$ can be expanded by Fourier series as

(4.8) \begin{equation} \begin{aligned} |\textrm {cos}\theta |\textrm {cos}\theta & = \sum _{n=0}^{\infty } \frac { \displaystyle \int _{0}^{2 \pi } |\textrm {cos}\theta |\textrm { cos}\theta \textrm {cos}n\theta \textrm {d}\theta }{\displaystyle \int _{0}^{2 \pi } \textrm {cos}^2 n\theta \textrm {d}\theta } \\ & = a_0 + a_1\textrm {cos}\theta + a_2\textrm {cos}2\theta + a_3\textrm {cos}3\theta + \ldots , \end{aligned} \end{equation}

where

(4.9) \begin{equation} a_n = 0 \; \textrm {for} \; n \; \textrm {is} \; \textrm {even}, \end{equation}

and

(4.10) \begin{equation} a_n = (-1)^{\frac {n+1}{2}}\frac {8}{n(n^2-4)\pi } \; \textrm {for} \; n \; \textrm {is} \; \textrm {old}, \end{equation}

therefore,

(4.11) \begin{equation} a_1 = \frac {8}{3\pi }, \; a_3 = \frac {8}{15\pi }, \; a_5 = -\frac {8}{105\pi }, \; \ldots . \end{equation}

Introducing the Fourier series expansion of term $|\textrm {cos}\theta |\textrm {cos}\theta$ , (4.8), into (4.5) and collecting the terms of the same trigonometric functions in the Fourier series expansion with the coefficients rearranged as

(4.12) \begin{equation} \left \{ \begin{aligned} B_1^{\prime } &= \frac {B_1}{a_1}, \\ B_3^{\prime } &= B_3 - B_1\frac {a_3}{a_1}, \\ B_5^{\prime } &= B_5 - B_1\frac {a_5}{a_1} \ldots , \end{aligned} \right . \end{equation}

we can have

(4.13) \begin{align} \frac {F_l}{\rho _{\!f} U_0^2 l} & = A_1\textrm {sin}\theta + A_3\textrm {sin}3\theta + A_5\textrm {sin}5\theta + \ldots \nonumber\\ &\quad + B_1^{\prime }|\textrm {cos}\theta |\textrm {cos}\theta + B_3^{\prime }\textrm {cos}3\theta + B_5^{\prime }\textrm {cos}5\theta + \ldots . \end{align}

Consequently, it can be written as

(4.14) \begin{align} \frac {C_l}{2} & = B_1^{\prime } + \frac {B_3^{\prime }\textrm {cos}3\theta }{|\textrm {cos}\theta |\textrm {cos}\theta } + \frac {B_5^{\prime }\textrm {cos}5\theta }{|\textrm {cos}\theta |\textrm {cos}\theta } + \ldots \nonumber\\ &\quad + \frac {A_1\textrm {sin}\theta }{|\textrm {cos}\theta |\textrm {cos}\theta } + \frac {A_3\textrm {sin}3\theta }{|\textrm {cos}\theta |\textrm {cos}\theta } + \frac {A_5\textrm {sin}5\theta }{|\textrm {cos}\theta |\textrm { cos}\theta } + \ldots . \end{align}

Since the lift coefficient is required to be phase-independent over a single oscillation period, all coefficients multiplying the trigonometric function terms must vanish. Hence, the lift coefficient of an inclined rigid plate in oscillatory flow can be expressed as

(4.15) \begin{equation} C_l = 2B_1^{\prime } = \frac {3}{4} \int _{0}^{2 \pi } \frac {F_l \textrm {cos} \theta }{\rho _{\!f} U_0^2 l} \textrm {d}\theta . \end{equation}

In figure 10(b), the lift coefficients are plotted as functions of the Keulegan–Carpenter number for different inclination angles $\alpha$ . The lift coefficient depends on $\alpha$ through both the exposed frontal area and the force decomposition induced by the inclination. Similar to the drag coefficient, the lift coefficient exhibits a scaling law with the Keulegan–Carpenter number as

(4.16) \begin{equation} C_l \sim KC^{-1/2}. \end{equation}

The lift coefficient, $C_l$ , initially increases with the inclination angle $\alpha$ , reaches a maximum at $\alpha = 45^{\circ }$ , and then decreases monotonically. Since the plate at $\alpha = 90^{\circ }$ has a symmetry shape with respect to the incoming flow, the resulting lift coefficient is negligibly small (close to zero) and therefore not suitable for presentation on a log–log plot. Accordingly, the data corresponding to $\alpha = 90^{\circ }$ are omitted from figure 10(b).

4.4. Plate deflection

In this section, we explain the influence of finite inclination angle on the plate response dynamics by examining the plate tip deflection amplitude, $A$ . The tip deflection is defined as the distance measured perpendicular to the initially undeformed plate and is calculated using the following expression:

(4.17) \begin{equation} A = \Delta x_{tip} \textrm {cos}(\pi /2 - \alpha ) + \Delta y_{tip} \textrm {sin}(\pi /2 - \alpha ), \end{equation}

where $\Delta x_{tip}$ and $\Delta y_{tip}$ are the horizontal and vertical displacements when the plate reaches the maximal deflection. Figure 11(a) shows the plate length-normalised tip deflection $A/l$ as a function of the reciprocal of the bending stiffness $K_{\!B}$ with varying ${\textit{KC}}$ and inclination angle $\alpha$ . In general, as the inclination angle increases, the plate undergoes more pronounced deflection due to the larger hydrodynamic forces acting on it. Similarly, as the bending stiffness decreases, the plate’s resistance to fluid loading from the ambient flow is weakened owing to its increasing structural flexibility, leading to a significant increase in tip deflection. However, when the bending stiffness becomes sufficiently small, the growth rate of the plate deflection is mitigated, which may be attributed to structural integrity constraints or the finite wave orbital excursion (Zhang & Nakamura Reference Zhang and Nakamura2024a ).

Figure 11.

Dimensionless tip deflections scaled by (a) plate length $l$ and (b) wave orbital excursion $a_w$ as a function of (a) $1/K_{\!B}$ and (b) ${\textit{CaL}}$ at various ${\textit{KC}} = 2\pi (\triangle )$ , $\pi (\bigcirc )$ and $\pi /2 (\square )$ and multiple inclination angles coloured as shown in the colour bar besides. The solid lines represent the scaling laws in (4.20).

The dependence of the tip deflection amplitude on bending stiffness is consistent with the vortex patterns shown in figures 8 and 9. At large bending stiffness, the plate undergoes negligible deformation, resulting in a large relative velocity between the structure and the oscillatory flow. The resulting large relative velocity promotes strong vortex shedding at both ends of the plate, particularly at higher inclination angles (figure 9 a). As the bending stiffness decreases, increased plate deflection modifies the local effective angle of attack, thereby altering both the onset and the spatial distribution of vortex shedding, as observed in figures 8 and 9.

The deflection process of the thin flexible plate in oscillatory flow is governed by the balance between the hydrodynamic forcing induced by the background flow and the restoring elastic force of the plate. The internal restoring moment by the elasticity of the plate is $EI \textrm {d}\theta / \textrm {d}s$ . By the observation of the deflected plate profile, the geometric relation can be approximated as $\theta \approx A/l$ and hence we can have $\textrm {d}\theta / \textrm {d}s \approx A/l^2$ . The moment by the hydrodynamic force acting perpendicularly on the undeformed plate can be divided into two components. The first is contributed by the drag force, $0.5C_d \rho _{\!f}U_0^2l^2b{\textrm {sin} \alpha }$ , and the lift force induced moment, $0.5C_l \rho _{\!f}U_0^2l^2b{\textrm {cos} \alpha }$ , is the second component. Therefore, the moment balance between the driven hydrodynamic force and the restoring elastic force can be written as

(4.18) \begin{equation} EI \frac {A}{l^2} \sim \frac {1}{2} C_d \rho _{\!f} U_0^2 l^2 b \textrm {sin} \alpha + \frac {1}{2} C_l \rho _{\!f} U_0^2 l^2 b \textrm {cos} \alpha , \end{equation}

where $b$ is the width of the plate exposed to the impinging flow and treated as unity in our two-dimensional configuration. If we combine the two terms from drag force and lift force in the right-hand side of the above equation, a modified Cauchy number can be derived as

(4.19) \begin{equation} Ca = (C_d \textrm {sin} \alpha + C_l \textrm {cos} \alpha ) \frac {\rho _{\!f} U_0^2 l^3}{EI}. \end{equation}

Therefore, the moment balance relation can be simplified as

(4.20) \begin{equation} \frac {A}{a_w} \sim \textit{CaL}, \end{equation}

where ${\textit{Ca}}$ is the modified Cauchy number above and $L$ is the ratio of plate length $l$ to wave orbital excursion $a_w$ . In figure 11(b), we plot the wave excursion-normalised tip deflection $A/a_w$ as a function of the product of ${\textit{Ca}}$ and $L$ . It is shown that our proposed scaling relation with the modified Cauchy number successfully predicts the plate deflection with varying structural elasticity and wave strength in the low value range, see the solid black line in figure 11(b). On the other hand, in the regime where ${\textit{CaL}}$ takes a relatively large value, the data points of wave excursion-normalised deflections scatter around the value of unity, as represented by the dash–dotted line. The distribution and variation of the results indicate that, similar to the trend of wall-mounted flexible plates in oscillatory flow (Zhang & Nakamura Reference Zhang and Nakamura2024a ), the tip deflection behaviour can be divided into two states: a fully reconfigured state, in which the balance between structural elasticity and fluid loading is achieved and the deflection follows $A/a_w \sim CaL$ law; and a passive movement state, where the tip deflection saturates to the wave orbital excursion, yielding $A/a_w \sim O(1)$ . Although some data points in the passive-movement regime exceed unity, this behaviour can be attributed to resonance when the modulated structural natural frequencies approach those of the oscillatory flow (Leclercq & de Langre Reference Leclercq and de Langre2018; Kumar, Kumar & Deepu Reference Kumar, Kumar and Deepu2021), with the modulation arising from the added-mass effect (Zhang & Nakamura Reference Zhang and Nakamura2024a ).

The plate tip deflection behaviour can be further elucidated from the perspective of energy transformation between the kinetic energy of the oscillatory flow and the mechanical energy of the thin flexible plate, which consists of motion-related kinetic energy and deformation-related elastic strain energy. The mechanical energy of the plate is extracted from the wave kinetic energy during each stroke through the work performed by the fluid loading on the plate. From the observation of temporal variation in plate motion and energy, it is noted that when the elastic strain energy of the plate reaches its maximum at the instant of greatest deformation, the plate comes to rest with zero velocity, resulting in vanishing kinetic energy. Therefore, here we represent the plate mechanical energy by its maximal elastic strain energy. We utilise a toy model of a torsional spring-loaded beam to evaluate the elastic strain energy of the deformed plate (Jerome et al. Reference Jerome, Bachelier, Doppler, Lehmann and Riviere2023). The parameter measuring the gained elastic strain energy with unity deformation is the effective stiffness $k$ , which can be approximated by the following equilibrium relation:

(4.21) \begin{equation} k\Delta \theta \sim \textit{EI} \frac {\textrm {d}\theta }{\textrm {d}s}, \end{equation}

where the plate angular deflection $\Delta \theta$ can be approximated as $\Delta x/l$ and consequently the plate curvature $\textrm {d}\theta / \textrm {d}s$ is estimated as $\Delta x/l^2$ . Therefore, the effective stiffness of the torsional spring-loaded beam is obtained in the form of

(4.22) \begin{equation} k \sim \frac {\textit{EI}}{l}. \end{equation}

The energy balance between the work done by the external fluid loading on the plate and the elastic strain energy of the plate deformation driven by the hydrodynamic force gives the equation

(4.23) \begin{equation} \frac {1}{2} k (\Delta \theta )^2 \sim \frac {1}{2} C_d \rho _{\!f} U_0^2 lbA \textrm {sin} \alpha + \frac {1}{2} C_l \rho _{\!f} U_0^2 lbA \textrm {cos} \alpha . \end{equation}

If we substitute the modified Cauchy number (4.19) and effective spring stiffness (4.22) into the energy balance relation above, the equation can be further simplified into the following form:

(4.24) \begin{equation} \frac {A}{a_w} \sim \textit{CaL}, \end{equation}

which is exactly the same as the scaling law derived from the moment balance.

As shown in figure 11, the successful prediction of the scaling law with the simulation results based on a modified Cauchy number, consisting of inclination angles $\alpha$ and inclination angle-dependent drag and lift force coefficients, $C_d$ and $C_l$ , suggests that the effect of the inclination angle on the kinematics and dynamics of the thin flexible plate can be interpreted from the following two complementary aspects. Geometrically, the inclination angle changes the projected area of the plate in its equilibrium configuration exposed to the flow. Mechanically, from the force perspective, the varying inclination between the plate and flow alters the instantaneous angle of attack, which further influences the fluid loading, causing the plate deflection. In addition, the modified Cauchy number is proven to reasonably reflect both the geometric and mechanical effects caused by the inclination angle, as evidenced by the good agreement between the scaling law prediction and simulation results.

4.5. Revisiting the phase diagrams

During the rightward stroke of the oscillatory flow, the fluid impinges on the leading edge of the inclined plate, which is analogous to the inverted-plate configuration in a uniform flow (Kim et al. Reference Kim, Cossé, Ceideira and Gharib2013; Gurugubelli & Jaiman Reference Gurugubelli and Jaiman2015). By contrast, during the leftward stroke, the flow encounters the fixed trailing edge and proceeds towards the free end, resembling a conventional plate in uniform flow (Zhang et al. Reference Zhang, Childress, Libchaber and Shelley2000; Connell & Yue Reference Connell and Yue2007). It is well established that the critical velocity for the onset of flapping in a conventional plate is significantly larger than that for an inverted plate (Kim et al. Reference Kim, Cossé, Ceideira and Gharib2013; Sader et al. Reference Sader, Cosse, Kim, Fan and Gharib2016a ). As a consequence, for an inclined plate subject to oscillatory flow, the onset of flapping is expected to be governed by a static divergence-type instability similar to that observed for inverted flexible plates (Kim et al. Reference Kim, Cossé, Ceideira and Gharib2013; Gurugubelli & Jaiman Reference Gurugubelli and Jaiman2015; Sader et al. Reference Sader, Cosse, Kim, Fan and Gharib2016a ).

Physically, this instability arises from the competition between the hydrodynamic loading and the elastic restoring force of the plate (Kim et al. Reference Kim, Cossé, Ceideira and Gharib2013; Gurugubelli & Jaiman Reference Gurugubelli and Jaiman2015; Sader et al. Reference Sader, Cosse, Kim, Fan and Gharib2016a ). For plates with a large spanwise aspect ratio, i.e. plate width much larger than plate length ( $b/l \gg 1.0$ ), Sader et al. (Reference Sader, Cosse, Kim, Fan and Gharib2016a ) showed that the critical condition for instability in uniform flow can be expressed in terms of a dimensionless flow speed $\kappa$ or Cauchy number ${\textit{Ca}}$ , with a critical value of approximately ${\textit{Ca}} = 1.85$ . However, the conventional definition of Cauchy number ${\textit{Ca}} = \rho _{\!f} U_0^2 l^3/EI$ does not fully characterise the hydrodynamic loading in oscillatory flow (Zhang & Nakamura Reference Zhang and Nakamura2024a ). In oscillatory flow, the hydrodynamic loading force acting on an inclined plate depends explicitly on the inclination angle $\alpha$ and the Keulegan–Carpenter number. These effects are reflected in the variation of the drag ( $C_d$ ) and lift ( $C_l$ ) coefficients with $\alpha$ and ${\textit{KC}}$ in the modified Cauchy number as proposed in the present study.

Accordingly, the critical condition for the transition from the static mode to the flapping mode can be expressed as follows:

(4.25) \begin{equation} \frac {C_d(\alpha ,\textit{KC}) \rho _{\!f} U_0^2 l^3 \textrm {sin} \alpha /2 + C_l(\alpha ,\textit{KC}) \rho _{\!f} U_0^2 l^3 \textrm {cos} \alpha /2}{EI} = C \end{equation}

where $C$ is a constant value. It can be further rearranged as

(4.26) \begin{equation} K_{\textit{Bcri}} = (C_d \textrm {sin} \alpha + C_l \textrm {cos} \alpha )/C .\end{equation}

We superimpose the prediction of (4.26) onto the phase diagrams in § 4.5 and find that it successfully captures the transition from the static mode to the flapping mode at ${\textit{KC}} = 2 \pi$ , ${\textit{KC}} = \pi$ and ${\textit{KC}} = \pi /2$ , as shown in figure 7. The critical conditions are plotted with the constant $C$ chosen as 0.07, 0.1 and 0.1414 for ${\textit{KC}} = 2 \pi$ , ${\textit{KC}} = \pi$ and ${\textit{KC}} = \pi /2$ , respectively. It is noted that both drag and lift coefficients follow the scaling $C_d \sim KC^{-1/2}$ and $C_l \sim KC^{-1/2}$ (see figure 10 a,b). The fitted constant $C$ varies with the Keulegan–Carpenter number, which implies that the critical bending stiffness $K_{\!B}$ is primarily governed by the geometrical factor, namely the inclination angle. This observation corroborates the hypothesis that the onset of flapping from the static mode is governed by a static divergence-type instability, which is irrelevant to the flow unsteadiness.

4.6. Drag reduction

In the interaction of fluid flow with flexible structures, once the external fluid loading exceeds a critical value, the structure undergoes a deformation compliant to the flow and even loses its stability, leading to flow-induced vibration (Boukor et al. Reference Boukor, Choimet, Laurendeau and Gosselin2018). Reciprocally, the reconfigured or unsteady structural geometry in turn affects the fluid loading experienced by the structure (Alben et al. Reference Alben, Shelley and Zhang2002; Boukor et al. Reference Boukor, Choimet, Laurendeau and Gosselin2018), which creates a strong coupling between them. In the preceding section, we discussed the structure deformation and response, i.e. plate tip deflection, with varying critical parameters. Here, we examine the effect of the inclination angle on the hydrodynamic drag force acting on a thin flexible plate with varying bending stiffness in oscillatory flow.

Figure 12.

Schematics of the profile superpositions of thin flexible plates in uniform incoming flow with inclination angle of (a) $90^{\circ }$ and (b) $0^{\circ }$ .

In figure 12, we present schematics of two extreme cases of thin flexible plates entering flapping regime under a uniform incoming flow with inclination angles of $0^{\circ }$ and $90^{\circ }$ . For the $90^\circ$ case, the thin flexible plate is placed vertically in a cross-flow, and the flow-induced reconfiguration reduces its frontal area and drag force (Luhar & Nepf Reference Luhar and Nepf2011). By contrast, when the thin flexible plate with zero inclination angle is positioned in the flow, theoretically, the form drag by pressure is negligible, while the skin friction drag plays a more important role. However, owing to the large-amplitude flapping of the thin flexible plate (see figure 12 b), it receives considerable pressure-induced drag force (Kim et al. Reference Kim, Cossé, Ceideira and Gharib2013). Therefore, these two extreme cases suggest that the inclination angle plays a crucial role in determining the drag force characteristics of the flexible plate, which is simultaneously dependent on the structure response behaviours governed by the competition between bending stiffness and fluid loading.

Figure 13.

The ratio of the dimensionless root-mean-square (r.m.s.) drag force of the inclined flexible plate to that of rigid one, $\widetilde {C_d} / \widetilde {C_{d,0}}$ as a function of $CaL$ with varying ${\textit{KC}}$ at (a) $\alpha = 5^{\circ }$ , (b) $\alpha = 30^{\circ }$ , (c) $\alpha = 60^{\circ }$ and (d) $\alpha = 90^{\circ }$ . The solid lines represent the scaling laws in (4.34). The dimensionless arithmetic mean drag force, $\overline {C_d}$ ,is shown in the corresponding insets.

To examine the influence of the bending stiffness and the inclination angle, in figure 13, we present the ratio of the dimensionless r.m.s. drag force of an inclined flexible plate to that of the corresponding rigid plate with varying $K_{\!B}$ number and Keulegan–Carpenter number for inclination angles $\alpha = 5^{\circ }$ , $30^{\circ }$ , $60^{\circ }$ and $90^{\circ }$ . In addition, the dimensionless arithmetic mean drag forces are shown in the inset figures. The r.m.s. drag forces of both flexible and rigid plates are calculated using the following equations:

(4.27) \begin{equation} \widetilde {F_d} = \sqrt {\frac {1}{T} \int _{0}^{T} F^2_d (t) \textrm {d} t}, \quad \textrm {and} \quad \widetilde {F_{d,0}} = \sqrt {\frac {1}{T} \int _{0}^{T} F^2_{d,0} (t) \textrm {d} t}, \end{equation}

where $F_d(t)$ and $F_{d,0}(t)$ denote the instantaneous drag force functions of the flexible and rigid plates, respectively, and $T$ is the period of the oscillatory flow. Accordingly, the ratio of the dimensionless r.m.s. drag force of the flexible plate to that of the rigid plate can be written as

(4.28) \begin{equation} \frac {\widetilde {C_d}}{\widetilde {C_{d,0}}} = \frac {\sqrt {\frac {1}{T} \int _{0}^{T} F^2_d (t) \textrm {d} t} / (\rho _{\!f} U_0^2 l)}{\sqrt {\frac {1}{T} \int _{0}^{T} F^2_{d,0} (t) \textrm {d} t} / (\rho _{\!f} U_0^2 l)} = \frac {\widetilde {F_d}}{\widetilde {F_{d,0}}}, \end{equation}

where $\widetilde {C_d}$ and $\widetilde {C_{d,0}}$ are the dimensionless r.m.s. drag forces. Meanwhile, the dimensionless arithmetic mean drag force, $\overline {C_d}$ , is calculated and normalised as

(4.29) \begin{equation} \overline {C_d} = \frac {1}{T} \int _{0}^{T} \frac {F_d(t)}{\rho _{\!f} U_0^2 l} \textrm {d} t. \end{equation}

In both mean drag force definitions, the mean values are averaged over 16 oscillation periods after the plate reaches the steady state for cases with regular responses, whereas for cases with irregular responses, the drag force is averaged over 40 periods.

In figure 13(a), it is observed that the flexible plate with a small inclination angle of $5^{\circ }$ exhibits drag forces essentially identical to those of a rigid plate over a wide range of ${\textit{CaL}}$ . Deviations occur only at very low stiffness, where irregular structural responses lead to increased drag force. On the other hand, for inclination angles $\alpha = 30^{\circ }$ , $60^{\circ }$ and $90^{\circ }$ in figure 13(b–d), the flexible plates exhibit closely similar trends in the variation of the drag ratio. When the combined parameter ${\textit{CaL}}$ is small ( ${\textit{CaL}} \lt 1$ ), indicating that fluid loading is significantly weak compared with the plate bending rigidity, the r.m.s. drag ratios are nearly independent of ${\textit{CaL}}$ and collapse onto the horizontal unity line. In this regime, the high structural bending stiffness forbids significant deflections, resulting in the plates experiencing the same drag force as the rigid ones. For some cases the drag ratio slightly exceeds unity, and similar behaviours are reported for a wall-mounted plate in oscillatory flow. This can be attributed to phase lag effects, which enhance the relative velocity between the plate and the oscillatory flow (Zhang & Nakamura Reference Zhang and Nakamura2024a ). From the perspective of vortex dynamics, for relatively stiff plates, a substantial fraction of the incoming flow kinetic energy is dissipated in the wake through strong shear layers (figure 8 a) and shed vortices (figure 9 a), resulting in high drag forces.

As the combined parameter ${\textit{CaL}}$ increases into the range of $O(1) - O(10)$ , a pronounced drag reduction is observed in figure 13(b–d), associated with increased structural reconfiguration. This enhanced reconfiguration arises from strengthened oscillatory flow (higher velocity and/or frequency) or reduced bending stiffness, both of which render the plate more deformable. Consequently, the effective projected area decreases due to reconfiguration, weakening the wake vortices and leading to a reduction in drag force. In this regime, the numerical results for the r.m.s. drag ratio collapse onto a straight line with a slope of $-1$ in a log–log representation. For larger values of ${\textit{CaL}}$ ( ${\textit{CaL}} \gt 10$ ), the r.m.s. drag ratio exhibits a slight but gradual increase followed by non-monotonic behaviour, which is attributed to the emergence of irregular flapping modes at very low bending stiffness.

To rationalise the observed trend of the r.m.s. drag force ratio for $\alpha = 30^{\circ }$ , $60^{\circ }$ and $90^{\circ }$ , we adopt the concept of the effective plate length introduced by Luhar & Nepf (Reference Luhar and Nepf2011) and Luhar & Nepf (Reference Luhar and Nepf2016). The effective plate length $l_e$ is defined as the length of a rigid plate that generates the same drag force as a flexible plate of total length $l$ , with both plates having the same inclination angle. This concept provides a convenient framework to quantify drag variations arising from frontal area reduction and streamlining effects induced by elastic reconfiguration. Under this formulation, the drag force can be evaluated using the same drag coefficient as that of a rigid plate. Accordingly, the ratio of r.m.s. drag forces can be expressed as

(4.30) \begin{equation} \frac {\widetilde {C_d}}{\widetilde {C_{d,0}}} = \frac {\widetilde {F_d}}{\widetilde {F_{d,0}}} = \frac {\widetilde {C_{d,0}} \rho _{\!f} U_0^2 l_e/2}{ \widetilde {C_{d,0}} \rho _{\!f} U_0^2 l/2} = \frac {l_e}{l} .\end{equation}

For small values of the combined parameter ${\textit{CaL}}$ ( ${\textit{CaL}} \lt 1$ ), the plate bending rigidity is sufficiently large that only negligible deflections occur under the oscillatory flow. The effective plate length therefore remains equal to the geometric length, i.e. $l_e / l = 1.0$ . As a result, the flexible and rigid plates experience essentially identical drag forces due to unaltered geometry features, and the r.m.s. drag ratio remains close to unity. On the other hand, in the parameter range where the reconfiguration-induced drag reduction becomes significant, the effective plate length variation can be elaborated by a scaling analysis of balance between the hydrodynamic fluid loading and the elastic restoring force of the structure. Under the external oscillatory flow, the flexible plate undergoes periodic flapping, back and forth, in one wave period, which gives rise to a tip deflection of the order of $O(a_w)$ , as shown in § 4.4. Similar to the derivation in § 4.4, the elastic restoring force of the flexible plate can be evaluated as

(4.31) \begin{equation} \textit{EI} \frac { {d}^2 \theta }{\textrm {d} s^2} \approx \textit{EI} \frac {a_w}{l^3}, \end{equation}

considering the geometry approximation $\theta \approx A/l \approx a_w/l$ . Following the definition of effective plate length, the balance between the hydrodynamic force exerted on the flexible plate and the elastic restoring force yields

(4.32) \begin{equation} \textit{EI} \frac {a_w}{l^3} \sim \frac {1}{2} C_d \rho _{\!f} U_0^2 l_e b \textrm {sin} \alpha + \frac {1}{2} C_l \rho _{\!f} U_0^2 l_e b \textrm {cos} \alpha \end{equation}

where $b$ is the width of the plate exposed to the impinging flow and treated as unity in our two-dimensional configuration, which further can be rewritten as

(4.33) \begin{equation} \frac {l_e}{l} \sim (\textit{CaL})^{-1}. \end{equation}

Indicated by (4.30), we can have the scaling results of r.m.s. drag force ratio as

(4.34) \begin{equation} \frac {\widetilde {C_d}}{\widetilde {C_{d,0}}} = \frac {l_e}{l} \sim (\textit{CaL})^{-1}, \end{equation}

of which the power of scaling law, $-1$ , agrees well with the numerical results in figure 13(b–d). The scaling results demonstrate that the observed drag reduction is primarily attributable to the decrease of effective plate length associated with the elastic reconfiguration of plates.

As shown in figure 13, it is also observed that the dimensionless arithmetic mean drag forces are generally close to zero, indicating that the instantaneous drag forces during the leftward and rightward strokes of the oscillatory flow are nearly symmetric. However, for moderate inclination angles (e.g. $\alpha = 30^{\circ }$ and $60^{\circ }$ ), $\overline {C_d}$ exhibits distinct positive peaks when ${\textit{CaL}} \sim O(1)$ . These peaks collapse well when they are scaled by the combined parameter ${\textit{CaL}}$ . A non-zero mean drag implies a net momentum input to the plate, which can be attributed to a shift in the equilibrium position about which the plate flaps. Specifically, under these conditions, the mean deflected configuration slightly deviates from the initial undeformed position, leading to a weak asymmetry in the force response over a wave cycle.

4.7. Energy conversion

A strong fluid–structure interaction is established between the thin flexible plate and the oscillatory flow. The hydrodynamic forces exerted on the plate surface alter the geometric configuration and may even affect its stability, which further changes fluid loading reciprocally. Through this coupled process, the ambient flow performs mechanical work on the reconfigured and flapping plate.

Figure 14.

Dimensionless averaged plate elastic strain energy as a function of ${\textit{CaL}}$ with varying ${\textit{KC}}$ number at (a) $\alpha = 5^{\circ }$ , (b) $\alpha = 30^{\circ }$ , (c) $\alpha = 60^{\circ }$ and (d) $\alpha = 90^{\circ }$ . The dash–dotted lines represent the zero value.

From an energetic perspective, this interaction can be interpreted as an energy transfer process. The work done by the fluid is converted into the kinetic energy of structural motion and the elastic strain energy associated with plate bending deformation. Subsequently, the elastic strain energy can be released into the kinetic energy of the vibrating plate, the vortex energy of shedding vortices and the viscous dissipation in the surrounding fluid. Overall, the energy transfer pathway proceeds from the kinetic energy of the oscillatory flow to the mechanical energy of the flapping plate.

In the context of harvesting renewable energy from environmental flows such as wind, water currents and waves, the elastic strain energy may be further converted into electrical energy using the techniques of piezoelectric materials (Orrego et al. Reference Orrego, Shoele, Ruas, Doran, Caggiano, Mittal and Kang2017; Hamlehdar, Kasaeian & Safaei Reference Hamlehdar, Kasaeian and Safaei2019) or triboelectric nanogenerator (Phan et al. Reference Phan, Shin, Jeon, Kang, Han, Kim, Kim, Kim, Hwang and Hong2017; Zhang et al. Reference Zhang, Meng, Xia, Deng, Dai, Hagedorn, Peng and Wang2020a ), enabling potential utilisations, such as providing power to small-scale offshore devices.

To explore the maximal potential of electricity power generation, we evaluate the elastic strain energy created by the structural deformation from the kinetic energy of the oscillatory flow, which is the foundation for electric energy utilisation. The evaluation is conducted by calculating the period-averaged elastic strain energy using the following expression:

(4.35) \begin{equation} E_s = \frac {1}{nT} \int _{t}^{t+nT} \int _{0}^{l} \frac {1}{2} EI \kappa ^2 \textrm {d}s\textrm {d}t, \end{equation}

where $\kappa$ is the local structural curvature, $EI$ is the bending stiffness and $T$ is the period of the oscillatory flow. The localised elastic strain energy is integrated along the plate and over multiple oscillatory periods. The period-averaged elastic strain energy is normalised by dynamic pressure $\rho _{\!f} U_0^2 l^2$ and denoted as $\overline {E_s}$ . Figure 14 presents the dimensionless period-averaged elastic strain energy $\overline {E_s}$ as a function of the combined parameter ${\textit{CaL}}$ for varying ${\textit{KC}}$ number at inclination angles $\alpha = 5^{\circ }$ , $30^{\circ }$ , $60^{\circ }$ and $90^{\circ }$ .

From figure 14, it is clearly observed that, for all inclination angles and Keulegan–Carpenter numbers, the dimensionless period-averaged elastic strain energy increase rapidly from zero value at very small ${\textit{CaL}}$ value until reaching a maximum value. Thereafter, $\overline {E_s}$ shows a swift decrease followed by the non-monotonic variation in the high ${\textit{CaL}}$ regime. At extremely low ${\textit{CaL}}$ , the plate remains nearly rigid and most of the fluid energy is dissipated in the wake, as shown in figures 8(a) and 9(a). The following increment of $\overline {E_s}$ can be explained by the fact that, at a fixed Keulegan–Carpenter number and inclination angle $\alpha$ , a decrease in $K_{\!B}$ , which by definition corresponds to an increase in ${\textit{CaL}}$ , renders the flexible plate less resistant to fluid loading. This promotes flapping with larger amplitudes and increased bending curvature. The combined effect of reduced bending stiffness and enhanced curvature gives rise to a continued growth of elastic strain energy. This is because the bending curvature $\kappa$ , which can be approximated as $\kappa \approx A/l^2$ as discussed in § 4.4, increases as the bending stiffness decreases in the fully reconfigured state. Moreover, the contribution of curvature growth dominates the effect of decreasing bending stiffness, since the elastic strain energy scales linearly with stiffness but quadratically with curvature, as indicated by (4.35). However, once the bending stiffness is sufficiently small for the plate to enter the passive movement state, the marginal contribution of curvature diminishes because the tip deflection saturates at the wave orbital excursion. Consequently, with nearly constant curvature and decreasing bending stiffness, the dimensionless elastic strain energy decreases. This rise-and-fall behaviour results in an optimal value of $\overline {E_s}$ , as shown in figure 14.

Around the optimal value, for example in the case shown in figure 9(b), the shed vortices are maintained in the vicinity of the plate leading edges, where they interact constructively with the plate, thereby promoting higher energy conversion. For lower bending stiffness, the plate becomes overly compliant to resist the shear imposed by the flow. Therefore, less fluid kinetic energy is transferred into the structural elastic strain energy, which is corroborated by the greatly weakened leading edge vortices observed in figure 9(c,d).

To further quantitatively elucidate the dependence of elastic strain energy on key parameters, namely the structural bending stiffness $K_{\!B}$ , the Keulegan–Carpenter number, and the inclination angle $\alpha$ , here we establish a scaling law model to show the influence of each aspect when the plate is in the fully reconfigured state. Based on the geometry of the deformed plate during bending, the radius of curvature can be approximated as

(4.36) \begin{equation} \frac {l}{R} = \theta \approx \frac {A}{l}, \end{equation}

which gives rise to a curvature as

(4.37) \begin{equation} \kappa = \frac {1}{R} \approx \frac {A}{l^2}. \end{equation}

According to the definitive expression of elastic strain energy of (4.35), the dimensionless elastic strain energy $\overline {E_s}$ scales as

(4.38) \begin{equation} \overline {E_s} \sim \frac {1}{T} \frac {EI}{\rho _{\!f} U_0^2 l^3} \frac {A^2}{l^4} l^2 T = \frac {EI}{\rho _{\!f} U_0^2 l^3} \frac {A^2}{a_w^2} \frac {a_w^2}{l^2}. \end{equation}

By substituting the definitions of dimensionless bending stiffness $K_{\!B} = EI/\rho _{\!f} U_0^2 l^3$ and length ratio $L = l/a_w$ , and applying the scaling law of tip deflection of plates in the fully reconfigured state, $A/a_w \sim CaL$ (4.20), we obtain the following scaling relation for $\overline {E_s}$ :

(4.39) \begin{equation} \overline {E_s} \sim {\textit{Ca}}^2K_{\!B}. \end{equation}

Figure 15.

(a) Dimensionless plate elastic strain energy as a function of ${\textit{CaL}}$ with varying Keulegan–Carpenter number and multiple inclination angles coloured as shown in the colour bar. The solid lines represent the scaling laws in (4.39). (b) Optimal Cauchy number ${\textit{Ca}}_{\textit{opt}}$ at which the flexible plate achieves the highest elastic strain energy from the oscillatory flow of various ${\textit{KC}}$ number under inclination angles of $\alpha = 30^{\circ }$ , $60^{\circ }$ and $90^{\circ }$ . The solid lines represent the scaling laws in (4.42).

In figure 15(a), we plot the dimensionless averaged elastic strain energy as a function of combined parameter ${\textit{Ca}}^2K_{\!B}$ for varying structural bending stiffness, Keulegan–Carpenter number and plate inclination angle. The proposed scaling law (4.39) is in good agreement with the simulation results in the fully reconfigured state. Although some data points exhibit deviations towards higher values, these discrepancies can be attributed to resonance-induced large amplitude flapping, which leads to a reduced radius of curvature and, consequently, enhanced elastic strain energy.

After the growing elastic strain energy reaches its maximum peak, $E_s$ decreases beyond a turning point. As shown in figure 15(a), the locations of these turning points, corresponding to the maximal elastic strain energy, vary with plate inclination angle under different oscillatory flow conditions. To identify the critical condition for optimal elastic strain energy, namely the maximum energy-harvesting potential, we hypothesise that this optimum occurs at the transition between the fully reconfigured state and the passive movement state, as suggested by the observations and discussions above.

Since $E_s$ increases quadratically with the plate tip deflection $A$ , larger deflections are particularly favourable for enhancing $E_s$ , despite its linear dependence on the bending stiffness $K_{\!B}$ . The state transition of a flexible plate in oscillatory flow occurs when the plate is able to reach the fully reconfigured state under the averaged flow velocity within one stroke prior to flow reversal. Two characteristic time scales for the reconfiguration procedures are therefore introduced: the time scale of oscillatory flow, $T_w = 1/f_w$ , and the time scale of flexible plate reconfiguration in a unidirectional uniform flow with an averaged velocity, $T_{\textit{uni}} \sim A /U_{aver}$ , where $A$ denotes the mean horizontal deflection of plate tip and $U_{aver}$ is the averaged velocity of uniform flow. For a sinusoidal oscillatory flow, the averaged wave velocity is represented by its r.m.s. value, $U_0/\sqrt {2}$ . Achieving full reconfiguration before the oscillatory flow reversal requires that the time when the flexible plate reaches the fully reconfigured shape be shorter than half of the oscillatory flow period, which leads to

(4.40) \begin{equation} T_{\textit{uni}} \lt T_{w}/2. \end{equation}

Therefore, the critical condition for optimal elastic strain energy is obtained by setting the above inequality to equality, as

(4.41) \begin{equation} T_{\textit{uni}} = T_{w}/2, \end{equation}

where $T_{\textit{uni}} = \sqrt {2} A/U_0$ and $T_w = 1/f_w$ . Previous studies on flexible plates subjected to a unidirectional uniform flow have shown that the tip deflection follows the scaling law $A/l \sim Ca$ (Jerome et al. Reference Jerome, Bachelier, Doppler, Lehmann and Riviere2023; Kumar et al. Reference Kumar, Assam and Prabhakaran2023). Therefore, (4.41) can be derived into the following equation:

(4.42) \begin{equation} {\textit{Ca}}_{\textit{opt}} \sim KC. \end{equation}

The critical Cauchy number ${\textit{Ca}}_{\textit{opt}}$ for the optimal energy conversion of elastic strain energy characterises the ability of a flexible plate to convert the work done by fluid loading into elastic strain energy. Figure 15(b) shows the optimal Cauchy number of inclined flexible plates with different inclination angles $\alpha$ under oscillatory flows with varying Keulegan–Carpenter number ${\textit{KC}}$ , where the data points corresponding to the maximal elastic strain energy are extracted from the simulation results in figure 14. The scaling law prediction derived from the time scale shows good agreement with the numerical results that all optimal Cauchy numbers of different inclination angles and wave orbital excursions collapse on a solid line. This collapse further confirms that the modified Cauchy number successfully captures the influence of inclination angle on the dynamics of inclined flexible plates in oscillatory flows.