3.1 Interaction of the plates with small separation, $s_{x}=0.5$

A first-order assessment of the plate motion as a function of $Ca$ is given with the time-averaged position of the tips $\boldsymbol{X}_{0}=(X_{0},Y_{0})=\langle x,y\rangle _{tip}$ in figure 2(a). The downstream plate (hereon  $P_{d}$ ) underwent lower mean deformations due to the locally reduced incoming flow. This is illustrated with the streamwise velocity distribution $U/U_{0}$ in figure 2(b,c) for two values of  $C_{a}$ ; there, streamlines are included to better visualize the mean flow features. Most of the $P_{d}$ was located within the back-flow region, i.e., $U(x,y)\leqslant 0$ , induced by the upstream plate (hereon  $P_{u}$ ). Only the vicinity of the tip was exposed to the nearly undisturbed flow. The relatively short gap between the plates also promoted the formation of two recirculation regions, which was not present with larger $s_{x}$ (shown later).

Figure 2.

(a) Time-averaged tip position ( $X_{0},Y_{0}$ ) of the upstream ( $P_{u}$ ) and downstream ( $P_{d}$ ) plates for $Ca\in [15,77]$ ; (b) streamwise velocity distribution $U/U_{0}$ at $Ca=15$ ; (c)  $U/U_{0}$ at $Ca=30$ . Cases with $s_{x}=0.5$ .

Figure 3.

(a) Intensity of the tip motions $I_{tip}$ across $Ca$ for the two plates; (b) cross-correlation function $\unicode[STIX]{x1D6FE}_{p_{u},p_{d}}(\unicode[STIX]{x1D70F})$ for selected $Ca$ ; (c) 2D compensated spectra of the tip displacements of the $P_{u}$ plate; (d) same as (c) for the $P_{d}$ plate. Cases with $s_{x}=0.5$ .

Bulk characterization of the dynamics of the plates was obtained via the intensity of the tip displacements $I_{tip}=\Vert \boldsymbol{x}(t)-\boldsymbol{X}_{0}\Vert /h$ , where $\Vert \cdot \Vert$ denotes the $L^{2}$ -norm and $\boldsymbol{x}$ is the instantaneous location of the tips. Notably, both plates exhibited very similar $I_{tip}$ , which increased monotonically with $Ca$ (see figure 3 a). The plates also showed a high degree of coupling, as evidenced by the cross-correlation of the tips $\unicode[STIX]{x1D6FE}_{p_{u},p_{d}}(\unicode[STIX]{x1D70F})=\langle x_{p_{u}}(t)x_{p_{d}}(t+\unicode[STIX]{x1D70F})\rangle$ in figure 3(b) for three $Ca$ . Indeed, peak values resulted in $\unicode[STIX]{x1D6FE}_{p_{u},p_{d}}(\unicode[STIX]{x1D70F}=t_{a})>0.65$ . The characteristic time shift $t_{a}$ is associated with the mean advection velocity of the flow between $P_{u}$ and $P_{d}$ and, consequently, decreased with increasing  $Ca$ .

The dominant modes of oscillation of the plates across $Ca$ are shown in figure 3(c,d) with the compensated spectra $f\unicode[STIX]{x1D6F7}/\text{max}\{\,f\unicode[STIX]{x1D6F7}\}$ , where $\unicode[STIX]{x1D6F7}$ is the spectrum of the streamwise component of the tip motions. In general, the upstream plate presented roughly similar features across $Ca$ with the dominating modes close to its first-order natural frequency $f_{n1}$ ; however, energetic modes at higher frequencies occurred in the downstream structure. The fundamental frequency $f_{n1}$ is modulated by the surrounding medium acting as added mass and damping; it is given by:

(3.1) $$\begin{eqnarray}f_{n1}/f_{n1o}=\sqrt{1-\unicode[STIX]{x1D709}^{2}},\quad \text{where}~f_{n1o}=\frac{0.56}{h^{2}}\sqrt{\frac{EI}{\unicode[STIX]{x1D70C}_{m}bc+\unicode[STIX]{x1D70C}_{f}C_{m}\unicode[STIX]{x03C0}b^{2}/4}}.\end{eqnarray}$$

Here, $C_{m}=1$ (Luhar & Nepf Reference Luhar and Nepf2016), $\unicode[STIX]{x1D709}$ is the damping ratio determined by the dominant oscillation frequency of a single plate under uniform flow at $Ca=15$ . Our experiments indicate that $\unicode[STIX]{x1D709}=0.43$ is a good estimation. Evidence indicates that for structures in tandem with sufficiently small separation, von Kármán vortex shedding can be induced only in the wake of the downstream structure (Lin & Glendening Reference Lin and Glendening2002; Lin, Yang & Rockwell Reference Lin, Yang and Rockwell2002; Borazjani & Sotiropoulos Reference Borazjani and Sotiropoulos2009; Younis, Alam & Zhou Reference Younis, Alam and Zhou2016). There, the gap may be dominated by small-scale Kelvin–Helmholtz ( $K{-}H$ ) instability (Khabbouchi et al. Reference Khabbouchi, Fellouah, Ferchichi and Guellouz2014; Liu, Hamed & Chamorro Reference Liu, Hamed and Chamorro2018). A similar phenomenon occurred with the plates for the $s_{x}=0.5$ cases, where the dynamics of $P_{u}$ was governed by $f_{n1}$ (Jin, Ji & Chamorro Reference Jin, Ji and Chamorro2016; Jin & Chamorro Reference Jin and Chamorro2017). Distinctive vortex shedding was induced in the very near wake of the downstream structure, whose footprint is observed in figure 3(d). The vortex shedding frequency ( $f_{v}$ ) did not strictly follow the Strouhal relation $St=f_{v}U_{0}/b=0.15$ (blue curve in figure 3 d) suggested by Knisely (Reference Knisely1990) and Norberg (Reference Norberg1993) for rigid plates. The differences in $St$ were likely due to the changes of the gap between the two plates across $Ca$ . Finally, it is worth pointing out that for sufficiently low $Ca$ ( ${\lesssim}30$ ), the $P_{d}$ was able to induce some backward modulation on  $P_{u}$ , as suggested in figure 3(c). This may be attributed to the comparatively smaller tip distance between the plates, shown in figure 2(a). The increase of $s_{x}$ is expected to severely condition the strong coupling of the structures and the turbulence-induced forcing on  $P_{d}$ ; this is explored as follows.

3.2 Interaction of the plates with intermediate separation, $s_{x}=1$

Figure 4.

Time-averaged tip position ( $X_{0},Y_{0}$ ) of the upstream ( $P_{u}$ ) and downstream ( $P_{d}$ ) plates for $Ca\in [15,77]$ ; (b) streamwise velocity distribution $U/U_{0}$ at $Ca=15$ ; (c)  $U/U_{0}$ at $Ca=30$ . Cases with $s_{x}=1$ .

The increase of the plate separation to $s_{x}=1$ or, equivalently, 4 times the plate’s width, led to very distinctive motions. Despite that $P_{u}$ exhibited similar levels of deformation across $Ca$ as those of the $s_{x}=0.5$ cases (figure 4 a), $P_{d}$ underwent smaller bending. This interesting phenomenon is explained by inspecting the streamwise velocity distribution shown in figure 4(b,c). Despite that the larger separation allowed a recovery of the bulk flow impinging $P_{d}$ , the flow in the vicinity of the $P_{d}$ tip was significantly lower (less than half of the freestream velocity) due to the vertical diffusion of the wake. This stresses the dominant role of the flow near the tip region. Note also that, compared to the $s_{x}=0.5$ cases, the velocity deficit distributed more uniformly at the location of  $P_{d}$ . Similar to cylinders in tandem (Lin & Glendening Reference Lin and Glendening2002; Younis et al. Reference Younis, Alam and Zhou2016), larger $s_{x}$ facilitates the production of vortices within the gap between the two plates; consequently, the $P_{d}$ structure is more susceptible to vortex-induced motions. This resulted in a large impact on $I_{tip}$ , $\unicode[STIX]{x1D6FE}_{p_{u},p_{d}}(\unicode[STIX]{x1D70F})$ and the dominant modes of oscillations. Indeed, the $I_{tip}$ of $P_{d}$ exhibited a distinctive trend compared to the $s_{x}=0.5$ cases, characterized by a concave distribution with a local minimum at intermediate $Ca$ (see figure 5 a); analysis of this phenomenon will be elaborated below. Also, loss of coupling between $P_{u}$ and $P_{d}$ is evident from a reduction in the peak of $\unicode[STIX]{x1D6FE}_{p_{u},p_{d}}(\unicode[STIX]{x1D70F})$ (see figure 5 b). The direct impingement of the vortex shedding from $P_{u}$ induced a strong footprint on the dynamics of $P_{d}$ approximately following $St=0.15$ (figure 5 d); there, the influence of tip vortex and bending of $P_{u}$ may lead to some other frequency components. Comparatively, the oscillations of $P_{u}$ presented a clear signature near $f_{n1}$ across $Ca$ (figure 5 c). As noted by Gopalkrishnan (Reference Gopalkrishnan1993) and Zhao (Reference Zhao2015), the effect of the von Kármán vortex shedding is usually more distinctive in the transverse direction (i.e., perpendicular to the mean flow) but comparatively reduced in the streamwise direction. A recent study by Jin, Kim & Chamorro (Reference Jin, Kim and Chamorro2018) shows that for flexible structure oscillations under uniform flow and restricted in streamwise direction, von Kármán vortex shedding dominates the wake of structures governed by natural-frequency motion; broadband flow fluctuations near wake play a distinctive role in driving the streamwise dynamics of the structure. Note, however, that larger amplitudes of oscillation of $P_{d}$ may eventually alter the dynamics of  $P_{u}$ , as noted in the compensated spectrum of the tip motion in figure 5(c) for high  $Ca$ .

Figure 5.

(a) Intensity of the tip motions $I_{tip}$ across $Ca$ for the two plates; (b) cross-correlation function $\unicode[STIX]{x1D6FE}_{p_{u},p_{d}}(\unicode[STIX]{x1D70F})$ for selected $Ca$ ; (c) 2D compensated spectra of the tip displacements of the $P_{u}$ plate; (d) same as (c) for the $P_{d}$ plate. Cases with $s_{x}=1$ .

Figure 6.

(a) Turbulence kinetic energy ( $TKE$ ) distribution for $Ca=15$ ; (b) same as (a) but for $Ca=30$ ; (c) average intensity of the velocity normal to the plate within a circular region centred at the tip with radius $R=\unicode[STIX]{x1D716}h$ for various  $\unicode[STIX]{x1D716}$ . Cases with $s_{x}=1$ .

The concave distribution of the tip vibration intensity deserves a close inspection due its impact on various natural and engineering systems. The dynamics of structures in the wake of other bodies is substantially modulated by the velocity fluctuations and the structure stiffness. In particular, comparatively higher turbulence kinetic energy $TKE=(1/2)\langle u^{\prime 2}+v^{\prime 2}\rangle /U_{0}^{2}$ in this two-plate system is located in the vicinity of the tip of $P_{d}$ (see two examples in figure 6 a,b). Here, the streamwise ( $u^{\prime }$ ) and vertical ( $v^{\prime }$ ) components of the velocity fluctuations may significantly affect the oscillations of  $P_{d}$ . If we consider that the dynamics of this plate is driven by the velocity fluctuations perpendicular to the tip vicinity $v_{n}$ (namely, normal to the tangent vector of the plate near the tip), a bulk effect is given by the standard deviation $\unicode[STIX]{x1D70E}_{vn}$ within a region around the tip; i.e., $\langle q\rangle \propto \unicode[STIX]{x1D70E}_{vn}^{2}$ , where $q$ is the fluctuating force. A region around the tip may be considered as a circular domain centred at the tip with radius $R=\unicode[STIX]{x1D716}h$ with $\unicode[STIX]{x1D716}\ll 1$ . Here, $\unicode[STIX]{x1D716}=0.1\pm 0.04$ showed a very consistent trend for estimating $\unicode[STIX]{x1D70E}_{vn}$ , which increased linearly with $Ca$ (see figure 6 c).

Figure 7.

(a) Local bending stiffness $\text{d}s^{\ast }/\text{d}p^{\ast }$ as a function of $X_{0}/h$ ; (b) driven-harmonic-oscillator frequency response function $\unicode[STIX]{x1D6FC}$ for the plate across $Ca$ ; (c) modelled and measured tip oscillation intensity $I_{tip}/\bar{I}_{tip}$ . Cases with $s_{x}=1$ .

Now, the model by Chen (Reference Chen2010) shows that for sufficiently large external forcing, a cantilever beam may undergo large deformation with strong nonlinear behaviour of the tip displacements. This model is used to estimate the differential tip displacement $\text{d}s$ as a result of a differential force $\text{d}p$ normal to the tip ( $\text{d}s/\text{d}p$ ) for a given mean streamwise deformation of the structure  $X_{0}$ . This is illustrated in figure 7(a) in non-dimensional form; i.e., $\text{d}s^{\ast }/\text{d}p^{\ast }=\text{d}s/\text{d}p(EI/h^{3})$ as a function of $X_{0}/h$ . Note that for small plate deformation $\text{d}s^{\ast }/\text{d}p^{\ast }\rightarrow 1/3$ , which agrees with that from the Euler–Bernoulli beam theory. It is also worth stressing the distinctive increase of $\text{d}s^{\ast }/\text{d}p^{\ast }$ for $X_{0}/h>0.5$ , which facilitates the occurrence of large-amplitude oscillations. As inferred from figure 5(d), the dynamics of $P_{d}$ is strongly modulated by the vortex shedding frequency from  $P_{u}$ . This modulation may be thought as a case of a driven harmonic oscillator (DHO); consequently, the shedding frequency defines the response function of the steady-sate solution of the DHO, as follows:

(3.2) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}=\frac{1}{\sqrt{(1-\unicode[STIX]{x1D6FE}^{2})^{2}+(2\unicode[STIX]{x1D709}\unicode[STIX]{x1D6FE})^{2}}},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}=f_{v}/f_{n1o}\approx StU_{0}b/f_{n1o}$ is the frequency ratio. Then, as $f_{v}$ departs from $f_{n1o}$ with increasing $Ca$ , the response function $\unicode[STIX]{x1D6FC}$ decreases monotonically (figure 7 b). Hence, the combined effect of the flow fluctuations, structure stiffness and dominant shedding frequency on $I_{tip}$ of $P_{d}$ can be approximated as:

(3.3) $$\begin{eqnarray}I_{tip|_{P_{d}}}\propto \unicode[STIX]{x1D70E}_{vn}^{2}\frac{\text{d}s^{\ast }}{\text{d}p^{\ast }}\unicode[STIX]{x1D6FC}.\end{eqnarray}$$

Figure 7(c) illustrates the estimated and measured $I_{tip}$ , normalized with the mean value $\bar{I}_{tip}$ across $Ca$ . It shows a good trend which explains the concave behaviour of $I_{tip}$ ; namely, for relatively low $Ca$ , $\text{d}I_{tip}/\text{d}Ca<0$ is governed by the comparatively fast reduction of $\unicode[STIX]{x1D6FC}$ , whereas for relatively high $Ca$ , $\text{d}I_{tip}/\text{d}Ca>0$ is modulated by nonlinear effects of the structure bending ( $\text{d}s^{\ast }/\text{d}p^{\ast }$ ).

Figure 8.

(a) Time-averaged tip position ( $X_{0},Y_{0}$ ) of the upstream ( $P_{u}$ ) and downstream ( $P_{d}$ ) plates for $Ca\in [15,77]$ ; (b) streamwise velocity distribution $U/U_{0}$ at $Ca=15$ ; (c)  $U/U_{0}$ at $Ca=30$ . Cases with $s_{x}=2$ .

Figure 9.

(a) Intensity of the tip motions $I_{tip}$ across $Ca$ for the two plates; (b) cross-correlation function $\unicode[STIX]{x1D6FE}_{p_{u},p_{d}}(\unicode[STIX]{x1D70F})$ for selected $Ca$ ; (c) 2D compensated spectra of the tip displacements of the $P_{u}$ plate; (d) same as (c) for the $P_{d}$ plate. Cases with $s_{x}=2$ .

Figure 10.

(a) Turbulence kinetic energy ( $TKE$ ) distribution for $Ca=15$ ; (b) same as (a) but $Ca=30$ ; (c) modelled and measured tip oscillation intensity $I_{tip}/\bar{I}_{tip}$ . Cases with $s_{x}=2$ .

3.3 Interaction of the plates with large separation, $s_{x}=2$

Increasing the plate separation to $s_{x}=2$ resulted in substantial recovery of the flow impinging  $P_{d}$ ; as a result, both plates exhibited minor differences in $\boldsymbol{X}_{0}$ across $Ca$ (see figure 8 a). Like the $s_{x}=1$ cases, the $P_{d}$ was fully within the wake (figure 8 b,c). In general, the distinctive trends of $I_{tip}$ for $P_{u}$ and $P_{d}$ are similar to those of the $s_{x}=1$ cases (figure 9). However, that for $P_{d}$ was lower at relatively high $Ca$ due to a reduced effect of the wake fluctuations $\unicode[STIX]{x1D70E}_{vn}$ near the tip. Indeed, distinctive differences of the $TKE$ near the tips are evident in figures 10(a) and 10(b); this phenomenon is also predicted by the model based on (3.3) (figure 10 c). The large separation equivalent to 8 times the plate width resulted in very weak coupling dynamics between $P_{u}$ and $P_{d}$ (figure 9 b). Like the other configurations, $P_{u}$ was dominated by frequencies close to $f_{n1}$ across $Ca$ (figure 9 c). This evidences that broadband flow fluctuation in the very near wake region of the flexible plate is more significant than the vortex shedding in streamwise-dominated oscillations. Comparatively, $P_{d}$ was modulated by both vortex shedding and the natural frequency (figure 9 d); this is possibly due to a weakening of the coherence of the vortex shedding in this relatively long separation compared to $s_{x}=1$ .