3.1 Aerodynamic force fluctuations and plate dynamics

Characterization of the plate motions and fluctuations of the aerodynamic force across $Ca$ and $\unicode[STIX]{x1D6FC}$ revealed distinctive patterns. First, quantification of the normal-force coefficient $C_{N}=2F_{N}/(\unicode[STIX]{x1D70C}_{f}U_{0}^{2}bL)$ is illustrated in figure 2(a); here, $F_{N}$ is the force component normal to the plate base. Despite the large variation of the inclination angle $\unicode[STIX]{x1D6FC}$ , the trends of $C_{N}$ as a function of $Ca$ are comparable. For $Ca\lesssim 20$ one has $C_{N}\propto Ca^{-1/3}$ and for $Ca\gtrsim 20$ one has $C_{N}\propto Ca^{-2/3}$ , corresponding to Vogel exponents of $V=-2/3$ and $-4/3$ . Similar results were noted by Gosselin, De Langre & Machado-Almeida (Reference Gosselin, De Langre and Machado-Almeida2010) on flexible rectangular plates (their figure 5), indicating that strong plate deformation may alter the interaction mechanism between the plate and flow. Bulk quantification of the plate dynamics can be obtained with the intensity of the force fluctuations. In particular, distributions of the component normal to the base of the plate $I_{N}$ ( $=2\unicode[STIX]{x1D70E}_{N}/(\unicode[STIX]{x1D70C}_{f}U_{0}^{2}bL)$ , where $\unicode[STIX]{x1D70E}_{N}$ is the standard deviation of $F_{N}$ ) as a function of $Ca$ are illustrated in figure 2(b). It shows negligible differences between plate inclinations for sufficiently low $Ca$ , and abrupt increase of the force fluctuations past a critical $Ca=Ca_{c}$ threshold that depended on the inclination angle. Such a critical condition occurred at $Ca_{c}=34$ and 22 for $\unicode[STIX]{x1D6FC}=30^{\circ }$ and $45^{\circ }$ cases, but it was absent at $\unicode[STIX]{x1D6FC}=0^{\circ }$ . To highlight the sharp change on the force fluctuations, figure 2(c,d) illustrates a sample time series of the normal force at the plate base $F_{N}$ for the case $\unicode[STIX]{x1D6FC}=45^{\circ }$ right before ( $Ca=20.5$ ) and after ( $Ca=24$ ) $Ca_{c}$ . Despite the similar averaged values, distinctive high-amplitude periodic oscillations occurred right within $Ca>Ca_{c}$ implying the onset of a different oscillation mode. This is also consistent with recent work by Leclercq, Peake & de Langre (Reference Leclercq, Peake and de Langre2018), where plate motions had a minor influence on the time-averaged flow-induced loads.

The instantaneous snapshots shown in figure 2(e–g) provide a basic linkage of the force fluctuation and distinctive dynamics of the plate motions for the particular case with $\unicode[STIX]{x1D6FC}=45^{\circ }$ . At $Ca<Ca_{c}$ (figure 2 e), the plate exhibited small-amplitude fluttering around the bending equilibrium. The increase of $Ca$ over $Ca_{c}$ (figure 2 f) resulted in twisting oscillations around a relatively larger bending. At sufficiently high $Ca$ , the plate tips exhibited comparatively large-scale orbital motions; this condition happened at $Ca\gtrsim 43$ ( $\unicode[STIX]{x1D6FC}=45^{\circ }$ ). There, the tip was dominated by anticlockwise orbits coupled with twisting oscillations, as illustrated in figure 2(g). See additional details in the supplementary movies available at https://doi.org/10.1017/jfm.2019.40.

The trajectories of the tip motions for various $Ca$ and $\unicode[STIX]{x1D6FC}$ illustrate the dominant modes of oscillation. Indeed, figure 3(a–c) shows minor fluttering corresponding to low $I_{N}$ around the bending equilibrium for $\unicode[STIX]{x1D6FC}=0^{\circ }$ . Such a mode is also observed under $\unicode[STIX]{x1D6FC}=30^{\circ }$ and $45^{\circ }$ within $Ca<Ca_{c}$ (figure 3 d,g). At $Ca_{c}$ , the plate experienced the onset of twisting oscillations. This resulted in stronger tip motions and, consequently, increased $I_{N}$ (figure 3 e,f,h,i). As expected, the intensity of tip oscillations increased with $Ca$ and $\unicode[STIX]{x1D6FC}$ . Dominant 3D orbital motions occurred at a specific threshold, which is the case in figure 3(i) at $Ca=68$ and $\unicode[STIX]{x1D6FC}=45^{\circ }$ . It is worth noting that such 3D dynamics led to significant deviations of the plate tips from the bending equilibrium; in this case the orbits from each tip overlapped.

Figure 3. Trajectories of the tips for various $\unicode[STIX]{x1D6FC}$ and $Ca$ : (a–c)  $\unicode[STIX]{x1D6FC}=0^{\circ }$ with $Ca=7$ , 41 and 68; (d–f) same as (a–c) with $\unicode[STIX]{x1D6FC}=30^{\circ }$ ; (g–i) same as (a–c) with $\unicode[STIX]{x1D6FC}=45^{\circ }$ . Here $x^{\ast }=x/L$ , $y^{\ast }=y/L$ and $z^{\ast }=z/L$ denote the normalized tip locations. The blue lines represent the location of the plate base.

Specific insight on the distinctive modes of oscillation is obtained by examining the representative trajectories and spectra of the tip velocity. In the fluttering-like mode, the bending deformation dominates the reconfiguration of the elastic plate where the tip is nearly parallel to the base. As noted in figure 4(a), the streamwise velocity spectrum $\unicode[STIX]{x1D6F7}^{\ast }$ of one of the tip edges evidences the modulation of the first- and second-order natural frequencies of the bending; these are given by $f_{b,r}=(j_{r}/L^{2})\sqrt{EI/(\unicode[STIX]{x1D70C}_{m}bc)}$ , where $j_{1}=0.56$ and $j_{2}=3.51$ . As reported in previous works (Fage & Johansen Reference Fage and Johansen1927; Modi et al. Reference Modi, Wiland, Dikshit and Yokomizo1992; Onoue et al. Reference Onoue, Song, Strom and Breuer2015; Liu et al. Reference Liu, Hamed, Jin and Chamorro2017), for a thin flat plate inclined with respect to the mean flow, the aerodynamic load produces a torque $M$ that tends to twist the frontal surface perpendicular to the free-stream flow. Its magnitude is given by

(3.1) $$\begin{eqnarray}M={\textstyle \frac{1}{2}}C_{M}(\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D70C}_{f}U_{e}^{2}Ab,\end{eqnarray}$$

where $C_{M}(\unicode[STIX]{x1D6FC})$ is the moment coefficient, $U_{e}$ is the effective wind speed and $A$ represents a characteristic area of the structure. An increase of the velocity results in an increment of the torque and the likelihood of tip twisting, which is controlled by the body stiffness. Such modulation, however, is continuously disturbed by the wake fluctuations resulting in twisting oscillations (figure 4 b). The height difference between the tips ( $\unicode[STIX]{x0394}z_{t}^{\ast }$ ) leads to the tip not being parallel to the base. In such cases, the tip velocity was substantially higher than the fluttering-like mode and exhibited semi-periodic variations. This is reflected by the dominating frequency component $f_{t}$ in the spectral domain. It is worth noting that $f_{t}$ remains nearly constant within all the twisting configurations, indicating that it is determined only by the characteristics of the structure.

Figure 4. Sampled tip trajectories and normalized streamwise velocity spectra $\unicode[STIX]{x1D6F7}^{\ast }=\unicode[STIX]{x1D6F7}/\text{max}\{\unicode[STIX]{x1D6F7}\}$ of a tip under (a) $Ca=7$ and $\unicode[STIX]{x1D6FC}=0^{\circ }$ , (b) $Ca=41$ and $\unicode[STIX]{x1D6FC}=30^{\circ }$ , (c)  $Ca=41$ and $\unicode[STIX]{x1D6FC}=45^{\circ }$ , and (d) $Ca=68$ and $\unicode[STIX]{x1D6FC}=45^{\circ }$ . The colour bar indicates the normalized height difference between the tips ( $\unicode[STIX]{x0394}z_{t}^{\ast }$ ). The two-headed blue arrow in (b) denotes the twisting oscillations, whereas the anticlockwise arrow in (d) indicates the 3D orbital motion of the tips composed of three stages. The insets show the (red) tip streamwise velocity  $u_{t}$ .

Figure 5. Normalized base normal-force spectra $\unicode[STIX]{x1D6F7}_{N}^{\ast }=\unicode[STIX]{x1D6F7}/\text{max}\{\unicode[STIX]{x1D6F7}\}$ at $\unicode[STIX]{x1D6FC}=45^{\circ }$ and (a) $Ca=36.3$ , (b) $Ca=38.6$ and (c) $Ca=41$ . The dashed arrows indicate the gradual increase of the frequency component corresponding to the orbital motions.

Figure 6. Instantaneous tip heights $z^{\ast }=z/L$ during one representative orbital cycle under $Ca=68$ and $\unicode[STIX]{x1D6FC}=45^{\circ }$ ; here $T_{r}=1/f_{r}$ is the period of orbital motion. The black and red points denote each tip.

Unlike the abrupt occurrence of the twisting mode from fluttering with $Ca$ , the transition from twisting to 3D orbits is comparatively a gradual process, where $I_{N}$ remains approximately constant. Indeed, the gradual development of the orbital motions may be noted from the changes on the normalized spectra $\unicode[STIX]{x1D6F7}_{N}^{\ast }$ of the base normal force $F_{N}$ with $Ca$ (figure 5). Within the transition between twisting and orbital modes, the twisting mode $f_{t}$ dominates the force fluctuations, whereas the signature of the orbital motions, $f_{r}$ , exhibits a monotonic increase with $Ca$ . In such a process, the force fluctuation intensity $\unicode[STIX]{x1D70E}_{N}^{2}=\int \unicode[STIX]{x1D6F7}\,\text{d}f$ does not show abrupt changes due to the very similar $\unicode[STIX]{x1D6F7}$ distributions across $Ca$ . The corresponding tip dynamics in the transition to orbital motions is illustrated with tip trajectories in figure 4(c) at $Ca=41$ and $\unicode[STIX]{x1D6FC}=45^{\circ }$ . It is characterized by twisting and 3D orbital motions. The associated tip velocity spectrum also reveals a dominant effect of the twisting mode ( $f_{t}$ ) and the orbital motion $f_{r}$ . The distinctive mode of orbital motions is observed only under $\unicode[STIX]{x1D6FC}=45^{\circ }$ and $Ca\gtrsim 43$ , which is a result of the coupling between plate twisting and bending as a three-stage process illustrated in figure 4(d) and figure 6 with instantaneous tip heights. The flow-induced torque at sufficiently high $Ca$ and $\unicode[STIX]{x1D6FC}$ leads to distinctive twisting of the plate near the tip such that the local projected area to the incoming flow increases (stage 1); this induces an abrupt increase of the aerodynamic load in the vicinity of the tip and substantial structure bending as highlighted by the decrease of $z^{\ast }$ of both tips (stage 2). As depicted in figure 2(f,ii), relatively high load can substantially deform the elastic plate; local bending angle of the plate near the tip relative to the vertical axis can reach or even be larger than $90^{\circ }$ . In this scenario (stage 3), the tip aerodynamic load drops significantly due to the reduced projected area. The structure restoring force governs the motion of the plate, pushing the tip back to the posture at stage 1, which completes an orbital cycle. Similar to the twisting mode, $f_{r}$ is independent of $Ca$ within the inspected configurations, despite the significant variation of the intensity of the orbital motions.

3.2 Theoretical estimation of the critical $Ca$ or $Ca_{c}$

An uncovered key quantity is the threshold $Ca_{c}$ that defines the condition of abrupt increase of force fluctuations leading to substantial unsteady loading. The aerodynamic load of an elastic structure undergoing simultaneous bending and twisting involves a number of physical processes. To tackle this problem quantitatively, we consider that the plate experiences minor deformation in a portion close to the tip. This assumption is well supported by the tracking of plate postures under $Ca<Ca_{c}$ (figure 2 d) and previous studies (Schouveiler, Eloy & Le Gal Reference Schouveiler, Eloy and Le Gal2005). Also, the plate dynamics shows that the plate remains nearly parallel to its base before the occurrence of the twisting mode. Hence, as a first-order approximation, we focus our analysis on the plate tip region with a constant local bending angle ( $\unicode[STIX]{x1D703}$ ) and inclination angle $\unicode[STIX]{x1D6FC}$ relative to the free-stream flow at equilibrium, as illustrated in figure 7(a). There, the governing equation of the twisting dynamics of the plate can be characterized by

(3.2) $$\begin{eqnarray}I_{w}\frac{\text{d}^{2}\hat{\unicode[STIX]{x1D6FC}}}{\text{d}t^{2}}+c_{w}\frac{\text{d}\hat{\unicode[STIX]{x1D6FC}}}{\text{d}t}+k_{w}\hat{\unicode[STIX]{x1D6FC}}=-\frac{1}{2}C_{M}\Big|_{\hat{\unicode[STIX]{x1D6FC}}_{e}}\unicode[STIX]{x1D70C}_{f}U_{e}^{2}Ab,\end{eqnarray}$$

where $\hat{\unicode[STIX]{x1D6FC}}$ is the modified inclination angle due to plate bending, $I_{w},~c_{w}$ and $k_{w}$ represent the polar moment of inertia, damping coefficient and torsional stiffness corresponding to the section under inspection, and $\hat{\unicode[STIX]{x1D6FC}}_{e}$ is the effective inclination angle modulated by the plate angular velocity. Here, the positive $M$ is defined to induce a decrease of $\hat{\unicode[STIX]{x1D6FC}}$ . For an inclined bent plate, the effective wind speed $U_{e}$ is taken as the velocity component normal to the plate span (figure 7 a), whereas the modified inclination angle at the equilibrium ( $\hat{\unicode[STIX]{x1D6FC}}_{s}$ ) is defined between $U_{e}$ and the normal vector $\boldsymbol{n}$ of the plate surface at the equilibrium bending position, as follows:

(3.3) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle U_{e}=U_{0}\sqrt{1-\cos ^{2}\unicode[STIX]{x1D6FC}\sin ^{2}\unicode[STIX]{x1D703}},\\[4.0pt] \displaystyle \sin \hat{\unicode[STIX]{x1D6FC}}_{s}=\frac{\sin \unicode[STIX]{x1D6FC}}{\sqrt{1-\cos ^{2}\unicode[STIX]{x1D6FC}\sin ^{2}\unicode[STIX]{x1D703}}}.\end{array}\right\}\end{eqnarray}$$

Figure 7. (a) Schematic illustrating parameters near plate tip region with bending angle $\unicode[STIX]{x1D703}$ . (b) Schematic of the cross-section of the plate undergoing twisting motion under the influence of instantaneous angular velocity. The ‘ $+$ ’ symbol denotes the plate centre. (c) Distribution of the plate tips projected in the $x{-}z$ plane under $Ca=7$ and (i)  $\unicode[STIX]{x1D6FC}=0^{\circ }$ , (ii)  $\unicode[STIX]{x1D6FC}=30^{\circ }$ and (iii)  $\unicode[STIX]{x1D6FC}=45^{\circ }$ . (d) Distribution of $C_{M}=2M/(\unicode[STIX]{x1D70C}_{f}U_{0}^{2}b^{2}L)$ with $\unicode[STIX]{x1D6FC}$ of a rigid plate with the same geometry.

The effective inclination angle $\hat{\unicode[STIX]{x1D6FC}}_{e}$ is influenced by the instantaneous angular velocity of the plate. Following the linear quasi-steady analysis characterizing the torsional instability (Païdoussis, Price & De Langre Reference Païdoussis, Price and De Langre2010), we consider an instant where the plate is in a given $\hat{\unicode[STIX]{x1D6FC}}$ with an instantaneous angular velocity $\text{d}\hat{\unicode[STIX]{x1D6FC}}/\text{d}t$ (figure 7 b). To represent the varying relative flow velocity along the width, we define a reference radius $b\unicode[STIX]{x1D709}$ such that the plate velocity is considered as $v=(\text{d}\hat{\unicode[STIX]{x1D6FC}}/\text{d}t)b\unicode[STIX]{x1D709}$ of the cross-section (Slater Reference Slater1969; Blevins Reference Blevins1990) and assumed to undergo minor variations within the investigated inclination angles (Tadrist et al. Reference Tadrist, Julio, Saudreau and De Langre2015). This leads to the relative wind velocity ( $\boldsymbol{U}_{rel}$ ) and effective inclination angle as

(3.4) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \boldsymbol{U}_{rel}=\boldsymbol{U}_{e}-\boldsymbol{v},\\[3.0pt] \displaystyle \hat{\unicode[STIX]{x1D6FC}}_{e}=\hat{\unicode[STIX]{x1D6FC}}-\unicode[STIX]{x1D6FE}.\end{array}\right\}\end{eqnarray}$$

For relatively small $\unicode[STIX]{x1D6FE}$ (small plate velocity compared to $U_{e}$ ) and low-amplitude twisting around the equilibrium,

(3.5) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle U_{rel}\simeq U_{e},\\ \displaystyle \unicode[STIX]{x1D6FE}\simeq \frac{v}{U_{e}}\sin \hat{\unicode[STIX]{x1D6FC}}\simeq \frac{v}{U_{e}}\sin \hat{\unicode[STIX]{x1D6FC}}_{s},\end{array}\right\}\end{eqnarray}$$

and, therefore,

(3.6) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D6FC}}_{e}=\hat{\unicode[STIX]{x1D6FC}}-\frac{\unicode[STIX]{x1D709}b}{U_{e}}\frac{\text{d}\hat{\unicode[STIX]{x1D6FC}}}{\text{d}t}\sin \hat{\unicode[STIX]{x1D6FC}}_{s}.\end{eqnarray}$$

Then, using Taylor’s expansion, the moment coefficient can be approximated as

(3.7) $$\begin{eqnarray}C_{M}\Big|_{\hat{\unicode[STIX]{x1D6FC}}_{e}}=C_{M}\Big|_{\hat{\unicode[STIX]{x1D6FC}}_{s}}+\left.\frac{\unicode[STIX]{x2202}C_{M}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}\right|_{\hat{\unicode[STIX]{x1D6FC}}_{s}}(\hat{\unicode[STIX]{x1D6FC}}_{e}-\hat{\unicode[STIX]{x1D6FC}}_{s}).\end{eqnarray}$$

The condition leading to the sudden increase of twisting motion denotes the loss of the system stability, which occurs when damping vanishes, i.e. the terms in $\text{d}\hat{\unicode[STIX]{x1D6FC}}/\text{d}t$ (combining (3.2), (3.6) and (3.7)) are zero:

(3.8) $$\begin{eqnarray}\left.\frac{\unicode[STIX]{x2202}C_{M}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}\right|_{\hat{\unicode[STIX]{x1D6FC}}_{s}}U_{e}\sin \hat{\unicode[STIX]{x1D6FC}}_{s}=\frac{2c_{w}}{\unicode[STIX]{x1D709}b^{2}\unicode[STIX]{x1D70C}_{f}A}=C,\end{eqnarray}$$

where $C$ is a constant. Key for evaluating the model is the estimation of the plate tip bending angle $\unicode[STIX]{x1D703}$ and, thus, $\hat{\unicode[STIX]{x1D6FC}}_{s}$ at each $Ca_{c}$ . Despite the variation of $\unicode[STIX]{x1D6FC}$ , the equilibrium tip height resulted very close at a given $Ca$ within $Ca<Ca_{c}$ (see example in figure 7 c), indicating similar plate bending; therefore, $\unicode[STIX]{x1D703}$ can be inferred from the case with $\unicode[STIX]{x1D6FC}=0$ at each $Ca_{c}$ directly from the PTV measurements. To determine $\unicode[STIX]{x2202}C_{M}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}|_{\hat{\unicode[STIX]{x1D6FC}}_{s}}$ , the aerodynamic torque of a rigid metallic plate with the same geometry of the elastic counterpart was measured across various $\unicode[STIX]{x1D6FC}$ as shown in figure 7(d), where the distribution of $C_{M}$ follows a power-law function with $\unicode[STIX]{x1D6FC}$ (see dashed line with power 1/2). Noting that $Ca\propto U_{0}^{2}$ for a given structure and combing (3.3) and (3.8), we can then define a dimensionless quantity $C_{t}$ characterizing the threshold of $Ca_{c}$ :

(3.9) $$\begin{eqnarray}C_{t}=\left.\frac{\unicode[STIX]{x2202}C_{M}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}}\right|_{\hat{\unicode[STIX]{x1D6FC}}_{s}}\sqrt{Ca_{c}}\sin \unicode[STIX]{x1D6FC}.\end{eqnarray}$$

Additional experiments were conducted to test equation (3.9), with the plate under various $\unicode[STIX]{x1D6FC}$ . As shown in figure 8, $Ca_{c}$ decreased monotonically with $\unicode[STIX]{x1D6FC}$ ; remarkably, $C_{t}\approx 0.2\pm 0.005$ across all investigated $\unicode[STIX]{x1D6FC}$ . It is worth pointing out that, compared to previous studies with rigid structures (Larsen Reference Larsen2002; Païdoussis et al. Reference Païdoussis, Price and De Langre2010; Fernandes & Armandei Reference Fernandes and Armandei2014), $\hat{\unicode[STIX]{x1D6FC}}_{s}$ accounts for the influence of plate flexibility on the onset of twisting motions. As indicated in (3.3), strong plate deformation (i.e. large bending angle $\unicode[STIX]{x1D703}$ ) leads to significant deviation of $\hat{\unicode[STIX]{x1D6FC}}_{s}$ from the base inclination angle $\unicode[STIX]{x1D6FC}$ , and therefore different local $\unicode[STIX]{x2202}C_{M}/\unicode[STIX]{x2202}\unicode[STIX]{x1D6FC}$ due to the nonlinear dependence of $C_{M}$ with $\unicode[STIX]{x1D6FC}$ (figure 7 d). Also, it should be noted that, for flexible plates with different geometry and material, the parameters on the right-hand side of (3.8) may vary and lead to different $C_{t}$ ; however, the critical $Ca_{c}$ should still follow (3.9) for a given structure. That is, the abrupt force fluctuations are triggered by instability mechanisms in an inclined bent plate relative to the mean flow.

Figure 8. $Ca_{c}$ and dimensionless quantity $C_{t}$ (3.9) characterizing the threshold of twisting motions; the dashed line marks the averaged value of $C_{t}$ across $\unicode[STIX]{x1D6FC}$ .