Flow-induced vibration of a flexible cantilever in tandem configuration

The unsteady three-dimensional Navier–Stokes equations are employed to predict the flow dynamics. Using an ALE reference frame on the fluid domain $\Omega^{\mathrm{f}}(t)$, the Navier–Stokes equations are written as:

where $\bm{u}^{\mathrm{f}}=\bm{u}^{\mathrm{f}}(\bm{x}^{\mathrm{f}},t)$ and $\bm{u}^{\mathrm{m}}=\bm{u}^{\mathrm{m}}(\bm{x}^{\mathrm{f}},t)$ are the fluid and mesh velocities related to every spatial point $\bm{x}^{\mathrm{f}}\in\Omega^{\mathrm{f}}$, respectively, $\bm{b}^{\mathrm{f}}$ represents the body force exerted on the fluid and $\bm{\sigma}^{\mathrm{f}}$ is the Cauchy stress tensor, given as:

where $p$ denotes the fluid pressure. The first term in Eq. 2 represents the partial derivative of $\bm{u}^{\mathrm{f}}$ with respect to time while the ALE referential coordinate $\hat{x}^{\mathrm{f}}$ is kept fixed. The fluid forcing acting on the cantilever’s surface is calculated by integrating the surface traction on the fluid-solid interface $\Gamma^{\mathrm{fs}}$. The instantaneous coefficients of the lift and drag forces are then computed as:

where $\bm{n}_{\mathrm{x}}$ and $\bm{n}_{\mathrm{y}}$ are the Cartesian components of the unit outward normal vector $\bm{n}$. A stabilized Petrov-Galerkin finite element method is employed to discretize the fluid domain $\Omega^{f}$ into $n_{el}$ non-overlapping finite elements $\Omega^{e}$ in space such that $\Omega^{f}=\bigcup_{e=1}^{n_{el}}\Omega^{e}$. The employed computational grid consists of unstructured prism elements, with a boundary layer mesh around the cylinders, as shown in Figs. 2b and 2c. The results for the grid convergence study are detailed in Sec. II.3.

We model the structural dynamics through a linear modal analysis by solving the Euler-Bernoulli beam equation. Let $\Omega^{\mathrm{s}}$ be the structural domain consisting of structure coordinates $\bm{x}^{\mathrm{s}}=(x,y,z)$. The transverse displacements $\bm{w}^{\mathrm{s}}(z,t)$ are solved using the Euler-Bernoulli beam equation excited by a distributed unsteady fluid loading per unit length $\bm{f}^{\mathrm{s}}$. Neglecting the damping and shear effects, the equation of motion of the beam is written as:

where $m=\rho^{\mathrm{s}}A$ is the mass per unit length of the beam, with $\rho^{\mathrm{s}}$ and $A$ being the density of the structure and the cross-sectional area, respectively. The Young’s modulus and second moment of area of the beam are denoted by $E$ and $I$, respectively. Considering the cantilever configuration, the boundary conditions at the ends of the beam are given as:

To find a solution for Eq. 6, we use a mode superposition approach, assuming small displacements from the mean position of the beam, characterized by the normal vibration modes. The frequency of the $\mathrm{i}^{\mathrm{th}}$ mode is given by:

in which $\lambda_{\mathrm{i}}$ is the nondimensional frequency parameter dependent on the mode number. For the current configuration, the eigenmodes are taken as the sums of sine, cosine, sinh, and cosh functions such that the eigenmode shape, denoted by $S_{\mathrm{i}}$, associated with the $\mathrm{{i}}^{\mathrm{th}}$ mode is written as:

where $\sigma_{\mathrm{i}}$ is a nondimensional parameter dependent on the mode number (see [34] for values of $\lambda_{\mathrm{i}}$ and $\sigma_{\mathrm{i}}$). Equation 6 can be rewritten into a matrix form as:

where $\bm{M}^{\mathrm{s}}$ and $\bm{K}^{\mathrm{s}}$ represent the mass and stiffness matrices, respectively, $\bm{w}^{\mathrm{s}}$ is the vector of unknown displacements across the span of the beam, and $\bm{f}^{\mathrm{s}}$ is the vector of the force acting on the beam. By projecting this equation into the eigenspace using the eigenmodes defined by Eq. 10, we transform Eq. 11 into a system of linear equations, with ’$\mathrm{i}$’ degrees of freedom, i.e, modes, as:

where ${\widetilde{\bm{M}^{\mathrm{s}}}}=\bm{S^{T}}\bm{M}^{\mathrm{s}}\bm{S}$ and ${\widetilde{\bm{K}^{\mathrm{s}}}}=\bm{S^{T}}\bm{K}^{\mathrm{s}}\bm{S}$ represent the projected matrices onto the eigenspace, with $\bm{S}$ representing the matrix containing the eigenvectors. The vector of the modal responses is denoted by $\bm{\zeta}^{\mathrm{s}}$. The relationship between the displacement vector $\bm{w}^{\mathrm{s}}$ and the vector $\bm{\zeta}^{\mathrm{s}}$ is given by $\bm{w}^{\mathrm{s}}=\bm{S\zeta}^{\mathrm{s}}$. The vector ${\widetilde{\bm{f}^{\mathrm{s}}}}$ in the above equation is the projected force vector given by ${\widetilde{\bm{f}^{\mathrm{s}}}}=\bm{S^{T}f}^{\mathrm{s}}$. The uncoupled Eq. 12 is solved for the modal amplitudes $\bm{\zeta}^{\mathrm{s}}$ which is then used to find the structural displacement. The structure is considered to be undeformed at time $t=0$. The modal amplitudes for the subsequent time steps are evaluated using the generalized-$\alpha$ time integration scheme [35, 36].

Let $\Gamma^{\mathrm{fs}}=\partial\Omega^{\mathrm{f}}(0)\cap\partial\Omega^{\mathrm{s}}$ be the fluid-solid interface at $t=0$ and $\Gamma^{\mathrm{fs}}(t)=\bm{\varphi}^{\mathrm{s}}(\Gamma^{\mathrm{fs}},t)$ be the interface at time $t$. We need to satisfy the continuity of velocity and traction at the fluid-solid interface, which requires satisfying the following conditions:

where, $\bm{\varphi}^{\mathrm{s}}$ represents the position vector that maps the initial position $\bm{x}^{\mathrm{s}}_{0}$ of the structure to its position at time $t$, i.e., $\bm{\varphi}^{\mathrm{s}}(\bm{x}^{\mathrm{s}},t)=\bm{x}^{\mathrm{s}}_{0}+\bm{w}^{\mathrm{s}}(\bm{x}^{\mathrm{s}},t)$, $\bm{t}^{\mathrm{s}}$ is the fluid traction vector corresponding to the fluid force as $\bm{f}^{\mathrm{s}}(z,t)=\int_{\Gamma^{\mathrm{fs}}}\bm{t}^{\mathrm{s}}\mathrm{d}\Gamma$, and $\bm{u}^{\mathrm{s}}$ represents the structure’s velocity at time $t$, defined as $\bm{u}^{\mathrm{s}}=\partial\bm{\varphi}^{\mathrm{s}}/\partial t$. Here, $\bm{n}$ denotes the outer normal to the interface, $\gamma$ represents any part of the interface $\Gamma^{\mathrm{fs}}$ in the reference configuration, $\mathrm{d\Gamma}$ is the differential surface area, and $\bm{\varphi}^{\mathrm{s}}(\gamma,t)$ corresponds to the fluid part at time $t$. These conditions ensure that the fluid velocity matches the velocity of the structure at the interface. To account for changes in the fluid-solid interface location, the motion of each spatial point in the fluid domain is explicitly controlled to ensure the kinematic consistency of the discretized interface. The spatial points $\bm{x}^{\mathrm{f}}\in{\Omega}^{\mathrm{f}}(t)$ are adjusted within the fluid domain by solving the following equation:

where $\bm{\sigma}^{\mathrm{m}}$ is the stress experienced by the spatial points due to the strain induced by the interface deformation. Assuming that the fluid mesh behaves as an elastic material, $\bm{\sigma}^{\mathrm{m}}$ could be expressed as:

where $k_{\mathrm{m}}$ is the local element-level mesh stiffness parameter, chosen based on element sizes, and $\bm{w}^{\mathrm{f}}$ represents the ALE mesh nodal displacement. This approach ensures the movement of internal finite element nodes maintains mesh quality, even under large structural displacements. Additionally, a Lagrangian finite element method is employed to adjust the mesh according to the updated domain geometry. A nonlinear partitioned iterative approach [37, 38] is used to couple the fluid and structure equations. At each time step, the fluid traction exerted on the structure’s surface is projected onto the eigenvectors to calculate the generalized modal forces. These projected modal forces are then used to determine the modal amplitudes and displacements for the subsequent time step. For a comprehensive review of the numerical algorithm and implementation details, interested readers are referred to [39].

We begin by examining the response characteristics of the flexible cantilever at a subcritical Reynolds number $Re=40$ and mass ratio $m^{*}=1$ for reduced velocities $U^{*}\in[1,19]$. We focus on characterizing the root-mean-square value of the nondimensional transverse oscillation amplitude $A_{\mathrm{y}}^{rms}/D$, along with the frequency of the oscillations and the fluid loading frequency, as functions of $U^{*}$. Figure 3a compares the amplitude of transverse oscillations across three distinct configurations: the isolated configuration, and the tandem configurations with the streamwise distance between the cylinders $x_{0}$ taken either as $x_{0}=5D$ or $x_{0}=10D$. In all three configurations, the cylinders have a similar circular cross-section of diameter $D$ and an aspect ratio $L/D=100$. According to Fig. 3a, the isolated cantilever exhibits a sustained oscillatory response for $U^{*}>5$. The amplitude of the transverse oscillation reaches a maximum value of $A_{\mathrm{y}}^{rms}/D\approx 0.50$ at $U^{*}=7$, gradually decreasing beyond this $U^{*}$.

For the tandem case with $x_{0}=5D$, the cantilever remains in its steady deflected position, i.e., no sustained oscillations, when $U^{*}\leq 7$. Within $U^{*}\in(7,16)$, the cantilever undergoes sustained oscillations, as depicted in Fig. 3a. In this configuration, the maximum transverse oscillation amplitude is marginally higher than the isolated case, reaching a value of $A_{\mathrm{y}}^{rms}/D\approx 0.55$ at $U^{*}=10$. Compared to the isolated configuration, the cantilever in the tandem cases experiences a sharp drop in the amplitude of oscillations beyond the peak point, and no sustained oscillations are observed for $U^{*}\geq 16$. The vibrational amplitudes for the tandem configuration with $x_{0}=10D$ remain relatively low across all examined reduced velocities, reaching a maximum value of $A_{\mathrm{y}}^{rms}/D\approx 0.20$ at $U^{*}=11$, as seen in Fig. 3a. In this configuration, sustained oscillations are present within $U^{*}\in(7,16)$, similar to the tandem case with $x_{0}=5D$.

The response curves presented in Fig. 3a indicate that the flexible cantilever in the tandem configuration is prone to sustained oscillations, similar to an isolated cantilever, at $Re=40$. The amplitude of transverse oscillations exhibits an inverse relationship with the spacing between the tandem cylinders, while the range of sustained oscillations remains within the same $U^{*}$ range for both tandem configurations. Figure 3b provides the values for the nondimensional transverse vibration frequency ($f_{\mathrm{y}}/f_{\mathrm{n}}$) and the nondimensional lift coefficient frequency ($f_{\mathrm{C_{L}}}/f_{\mathrm{n}}$) as functions of $U^{*}$. In all the instances where the flexible cantilever exhibits sustained oscillations, a frequency match is observed between the frequency of the transverse oscillation and the frequency of the lift coefficient. A typical lock-in behavior is found to be present during the cantilever’s oscillatory response. To gain a more comprehensive understanding of the cantilever’s response dynamics, we have presented the time histories of the nondimensional transverse displacement ($(y-\overline{y})/{D}$) and the cross-sectional lift coefficient ($C_{l}-\overline{C_{l}}$) at the free end of the cantilever for the tandem configuration with $x_{0}=5D$ at $U^{*}=9$ in Fig. 4a. According to the time series plot, the transverse displacement exhibits a time-periodic behavior in phase with the lift coefficient. For all the studied cases, we find that there is a match between the frequency of the transverse oscillation and the frequency of the cross-sectional lift coefficient across the length of the cantilever, as shown in Fig. 4b for the representative tandem case at $U^{*}=9$.

Next, we investigate the spatio-temporal response of the cantilever and the fluid loading coefficients in both the streamwise and transverse directions along the cantilever length. The results for the representative tandem case at $U^{*}=9$ are illustrated in Fig. 5. Figures 5a and  5b present the variations of the cross-sectional drag and lift coefficients, respectively, in the time domain. According to Figs. 5a and  5b, the variations in the lift coefficient are more pronounced than the drag coefficient, reflecting the stronger influence of transverse fluid forces on the cantilever’s oscillatory response. The variations of the nondimensional streamwise and transverse displacements in the time domain across the length of the cantilever are shown in Figs. 5c and  5d, respectively. The streamwise displacement shows small amplitude oscillations, reaching peak magnitudes of about $0.01D$, while the transverse displacement exhibits significant oscillations, with peak magnitudes of approximately $0.75D$. The larger amplitude of the transverse displacement compared to the streamwise displacement indicates that the oscillatory motion along the cantilever’s length is primarily in the transverse direction, with the lift force exerting a significant influence on the cantilever’s response.

An isometric view of the cantilever’s motion trajectory in the tandem configuration with $x_{0}=5D$ at $U^{*}=9$ is illustrated in Fig. 6a. A figure-eight motion trajectory is observed along the cantilever’s length, characterized by dominant vibration amplitudes in the transverse direction. This pattern is consistent across all cases where the cantilever exhibits sustained oscillations. In these instances, the streamwise vibration frequency is twice the transverse vibration frequency along the cantilever’s length, resulting in the observed figure-eight shape trajectories.

Next, we examine the power transfer between the transverse fluid loading and the flexible cantilever to characterize the energy exchange mechanisms driving the system’s dominant oscillatory behavior in the transverse direction. Figure 7 illustrates the time history of the instantaneous power transfer from the fluid flow to the flexible cantilever over a single period of its transverse oscillation ($T$) for both isolated and tandem configurations at $U^{*}=9$. The nondimensional power transfer coefficient in the transverse direction is defined as $(C_{\mathrm{l}}-\overline{C_{\mathrm{l}}})v/U_{0}$, where $v$ is the cross-sectional transverse velocity of the structure. A time-periodic interaction between the flexible cantilever and the fluid loading is observed, resulting in a cyclic power transfer in and out of the flexible structure. Given the fluid loading and the cantilever’s vibration response, the hydrodynamic coefficient in phase with the cantilever’s transverse velocity is calculated using the following equation:

where $C_{\mathrm{lv}}$ represents the hydrodynamic coefficient in phase with the transverse velocity of the structure. Figure 7d illustrates the distribution of $C_{\mathrm{lv}}$ along the cantilever length for the isolated and tandem configurations at $U^{*}=9$. We assume that $C_{\mathrm{lv}}=0$ at the fixed end of the cantilever. As shown in Fig. 7d, both the isolated configuration and the tandem case with $x_{0}=5D$ exhibit significant variations in $C_{\mathrm{lv}}$ values. Notably, relatively large positive $C_{\mathrm{lv}}$ values are observed in regions extending up to approximately 80% of the cantilever length. These large positive values enhance energy transfer from the fluid to the solid, contributing to the sustenance of oscillations. In the following, we provide insights into the vorticity dynamics and discuss how wake structures evolve and interact with the flexible cantilever during its oscillatory motion.

The wake dynamics of the flexible cantilever in tandem configuration within the subcritical Reynolds number regime reveal complex fluid-structure interactions that differ from those of an isolated cantilever. At a representative reduced velocity $U^{*}=11$, the $z$-vorticity ($\omega_{\mathrm{z}}$) contours at the cantilever’s free end illustrate the profound impact of the upstream cylinder on the wake structure. In the isolated configuration, we observe a classic von Kármán vortex street, characterized by alternating positive and negative vorticity regions. This pattern is indicative of strong transverse oscillations, wherein the periodic shedding of vortices creates an oscillating lift force that excites the cantilever’s motion. However, in the tandem configuration, there is a critical modification to this behavior. As shown in Fig. 8, the upstream cylinder’s presence delays vortex formation behind the cantilever, resulting in an extended attached near-wake region. This delay is a consequence of the upstream wake interfering with the shear layers separating from the cantilever, altering the momentum transfer in the boundary layer and postponing the roll-up of vortices. The extended near-wake in tandem configuration significantly impacts the coupled system’s dynamics. Specifically, the frequency of the lift force shifts to lower values in the tandem configuration compared to the isolated case, as evidenced previously in Fig. 3b.

Examining the $z$-vorticity contours along the cantilever’s length for the tandem configuration with $x_{0}=5D$ at a representative $U^{*}=11$ reveals a spatial variation in flow stability, as shown in Fig. 9, that is crucial for understanding the system’s behavior. At the fixed end, i.e., $z/L=0$, we observe a steady wake behind the cantilever, indicating dominant viscous forces and weak fluid-structure coupling. As we progress towards the free end, i.e., $z/L=1$, the wake transitions to an unsteady state with periodic vortex shedding downstream. This spatial variation in wake stability is a result of the cantilever’s varying flexibility and vibration amplitude along its length. The free end, experiencing larger oscillations, induces stronger shear layer instabilities, promoting vortex formation even within the laminar subcritical regime of $Re$. The three-dimensional wake structures, visualized through $z$-vorticity iso-surfaces, provide further insight into the coupled system’s behavior across different $U^{*}$ regimes. As shown in Fig. 10, in the pre-lock-in regime ($U^{*}=7$), the wake remains largely steady, indicating that the kinetic energy of the flow is insufficient to overcome viscous damping and induce unsteady vortex shedding. The lock-in regime ($U^{*}=11$) exhibits well-defined periodic vortex shedding patterns, with a strong synchronization process that facilitates maximum energy transfer from the fluid to the structure, resulting in sustained large amplitude oscillations. In the post-lock-in regime ($U^{*}=19$), the wake reverts to a steady state. These observations lead us to identify the critical conditions for sustaining wake unsteadiness and oscillations in the tandem configuration at the laminar subcritical regime of $Re$: (i) Sufficient flow inertia to overcome viscous damping and develop instabilities in the shear layer and (ii) system parameters within the lock-in range to allow for sustained energy transfer and overcome the coupled system’s damping mechanisms. In the following, we further investigate how system parameters influence the wake dynamics and oscillatory response of the cantilever by studying the coupled dynamics in the post-critical regime of $Re$.

Here, we provide a detailed comparison of the cantilever’s amplitude response and frequency dynamics in the isolated and tandem configurations at two post-critical Reynolds numbers of $Re=60$ and 100 for $U^{*}\in[1,19]$ at $m^{*}=1$. We focus on a single representative separation distance $x_{0}=5D$ for the tandem configuration. Figure 11a presents the values of $A_{\mathrm{y}}^{rms}/D$ for the isolated and tandem cases at $Re=60$. In the isolated configuration, the amplitude of transverse oscillations is close to zero for $U^{*}\leq 4$. The oscillations reach a peak amplitude at $U^{*}=7$, with a value of $A_{\mathrm{y}}^{rms}/D\approx 0.55$. Beyond $U^{*}=7$, there is a gradual decrease in the amplitude of oscillations, similar to the isolated case at $Re=40$. In the case of the cantilever in tandem configuration, the transverse oscillations reach a peak amplitude at $U^{*}=9$, with a value of $A_{\mathrm{y}}^{rms}/D\approx 0.80$. Compared to the tandem cases at $Re=40$ where no sustained oscillations were present in the post-lock-in regime, for the tandem cases at $Re=60$, the cantilever undergoes sustained oscillations for $U^{*}\geq 15$, with values of $A_{\mathrm{y}}^{rms}/D\approx 0.22$, as depicted in Fig. 11a.

Figure 11b illustrates the nondimensional transverse vibration frequency $f_{\mathrm{y}}/f_{\mathrm{n}}$ at the free end of the cantilever and the nondimensional frequency of the lift coefficient $f_{\mathrm{C_{L}}}/f_{\mathrm{n}}$ as functions of $U^{*}$ for the isolated and tandem configurations at $Re=60$. In both cases, there is a match between the frequency of the transverse oscillations and the frequency of the lift coefficient for all studied $U^{*}$. For the isolated case, the transverse oscillation and lift coefficient frequencies closely match the cantilever’s first-mode natural frequency within the range $U^{*}\in(5,8)$, resulting in peak vibration amplitudes in this range. In the tandem configuration, this range of peak amplitude response is shifted to $U^{*}\in(7,9)$. Based on the results provided in Fig. 11, it is evident that the flexible cantilever in the tandem configuration exhibits sustained oscillations similar to those of an isolated flexible cantilever experiencing VIVs.

Next, we compare the values of $A_{\mathrm{{y}}}^{rms}/D$ for the isolated and tandem configurations at $Re=100$. As shown in Fig. 12a, in the isolated configuration, the amplitude of the transverse oscillation follows a bell-shaped curve as a function of $U^{*}$. Similar to the isolated case at $Re=60$, the oscillations reach a peak amplitude at $U^{*}=7$, with the value of $A_{\mathrm{y}}^{rms}/D\approx 0.60$. For the cases in the tandem configuration, the transverse oscillation reaches a peak value at $U^{*}=7$, with a value of $A_{\mathrm{{y}}}^{rms}/D\approx 0.80$. This peak in the value of the transverse oscillation amplitude is similar to that observed in the tandem configuration at $Re=60$. Compared to the tandem cases at $Re=60$, the flexible cantilever at $Re=100$ experiences lower amplitude oscillations for $U^{*}\geq 15$, with the value of $A_{\mathrm{{y}}}^{rms}/D\approx 0.10$.

The values of $f_{\mathrm{y}}/f_{\mathrm{n}}$ and $f_{\mathrm{{C_{L}}}}/f_{\mathrm{n}}$ in terms of $U^{*}$ for the isolated and tandem cases at $Re=100$ are provided in Fig. 12b. In the isolated configuration, the cantilever experiences a single-frequency response, where there is a match between the frequency of the transverse oscillation and the frequency of the lift coefficient across all examined $U^{*}$. Compared to the tandem cases at $Re=60$, a noticeable secondary frequency component is present in the frequency of the transverse oscillation and lift coefficient frequency at $Re=100$. Specifically, we find that within $U^{*}\in[9,17]$, the dominant frequency of the transverse oscillation remains close to the cantilever’s first-mode natural frequency, while the secondary frequency component of the transverse oscillation matches the vortex-shedding frequency of a rigid stationary cylinder at $Re=100$, as illustrated in Fig. 12b. These observations indicate that for the flexible cantilever in the tandem configuration, as $Re$ increases, i.e., increasing inertial dominance in the flow, the cantilever becomes more susceptible to a multi-frequency response at high $U^{*}$ values and tends to oscillate predominantly at frequencies close to its first-mode natural frequency over a wider range of $U^{*}$.

Based on the results provided in Fig. 12, the response dynamics of the cantilever in the tandem configuration at $Re=100$ could be classified into two main categories in terms of $U^{*}$; VIV-dominated and WIV-dominated regimes. For $U^{*}<8$, the phenomenon underlying the cantilever’s oscillatory response is similar to that of an isolated cantilever. As shown in Fig. 12b, a single frequency component is observed in the transverse force, which is induced by vortex shedding from the cantilever and synchronizes with the transverse oscillation frequency. Consequently, this vibration regime is characterized by a VIV process, where the vortex shedding frequency matches the cantilever’s oscillation frequency. For $U^{*}\in[9,17]$, two dominant frequencies exist in the transverse fluid loading and the oscillatory response, as shown in Fig. 12b. The oscillatory response is primarily driven by the low-frequency component of the transverse load, which approximates the cantilever’s first-mode natural frequency. Notably, the low-frequency component in the transverse loading is approximately half the vortex-shedding frequency of the upstream cylinder. Within this $U^{*}$ range, the transverse oscillation frequency exceeds the cantilever’s first-mode natural frequency but remains below the vortex shedding frequency.

To highlight the differences between the cantilever’s oscillatory response and the transverse fluid forces in the tandem configuration for the VIV- and WIV-dominated regimes, we have provided the time history of $C_{l}-\overline{C_{l}}$ and $(y-\overline{y})/{D}$ at the free end of the cantilever at $U^{*}=5$, representative of VIV-dominated regime, and $U^{*}=9$, representing the WIV-dominated regime, in Fig. 13. The results presented in Fig. 13 further illustrate the cantilever’s multi-frequency response in the WIV-dominated regime at $Re=100$, compared to the VIV-dominated range, where the oscillations exhibit a single-frequency response. For the tandem cases at $Re=100$, when $U^{*}\in[18,19]$, the response dynamics and the transverse fluid forces converge to a single-frequency response. The presented results suggest that at higher reduced velocities than those investigated in this study, the cantilever could exhibit sustained oscillations at its higher harmonic natural vibration modes, as experimentally observed in flexible cylinders in tandem arrangement [40]. While a comprehensive examination of these vibrations is beyond the scope of our current work, it represents a promising direction for future investigations. In the following, we focus on the wake interference and its impact on the cantilever’s oscillatory response in the post-critical $Re$ regime.

Next, we present the vorticity dynamics for the isolated and tandem configurations in the post-critical Reynolds number regime. Figure 14 provides an isometric view of the $z$-plane slices of the $z$-vorticity contour at $Re=60$ for the tandem arrangement at a representative $U^{*}=11$. We observe that a fully developed vortex street is present at the free end, i.e., $z/L=1$. Progressing to the mid-span, i.e., $z/L=0.5$, the vortex street remains evident with a gradual modification of the wake structure. At the fixed end, i.e., $z/L=0$, a notable transformation in the wake structure is observed, wherein the flow pattern transitions to elongated, parallel vorticity layers. This quasi-steady flow regime near the cantilever’s fixed end signifies the influence of the cantilever’s motion and flow inertia in determining wake dynamics in the tandem cylinder arrangement. Specifically, we observe that for an isolated flexible cantilever under similar conditions, the flow exhibits unsteady dynamics characterized by a mature von Kármán vortex street along the entire cantilever length. However, in the tandem arrangement, the wake structures exhibit spatial variations: a well-defined vortex street at the free end and a more stable flow near the fixed end.

To compare the wake dynamics between the isolated and tandem configurations, we have presented the $z$-plane slices of the $z$-vorticity contour at the mid-span of the cantilever at $Re=60$ for three distinct reduced velocities $U^{*}=4$, 7, and 14 in Fig. 15. At $U^{*}=4$, the wake of the cantilever in the isolated configuration exhibits unsteady dynamics with clear vortex-shedding patterns, while the tandem configuration shows a nearly steady, streamlined flow with no distinct vortex shedding. The presence of the upstream cylinder delays vortex formation and roll-up, leading to a more stable flow regime compared to the isolated case. As the reduced velocity increases, we observe the onset of wake instability and asymmetry in the tandem arrangement. At $U^{*}=7$ and 14, both configurations exhibit unsteady wakes with vortex-shedding patterns, as shown in Fig. 15.

A similar comparison between the wake dynamics of the isolated and tandem configurations is presented in Fig. 16 for $U^{*}=4$, 7, and 14 at $Re=100$. For all the studied $U^{*}$ values, we observe unsteady vortex-shedding patterns in both isolated and tandem configurations. At $Re=100$ and $U^{*}=4$, the tandem configuration shows periodic vortex shedding, contrasting with the steady wake observed at $Re=60$ at the same $U^{*}$. The higher Reynolds number amplifies flow instabilities, resulting in vortex shedding even at lower reduced velocities. As $U^{*}$ increases, we see intensified vortex merging and splitting in the tandem arrangement, as shown in Fig. 16.

The presented results underscore the critical roles of both the Reynolds number $Re$ and reduced velocity $U^{*}$ in determining the wake dynamics in the tandem configuration. At lower values of $Re$ and $U^{*}$, such as in the tandem configuration at $Re=60$ and $U^{*}=4$, the wake remains nearly steady and symmetric, indicating minimal disturbances in the flow. Conversely, as $Re$ and $U^{*}$ increase, the flow becomes significantly more unstable. Higher $Re$ values enhance flow instabilities, while increased $U^{*}$ adds to the wake complexity by intensifying the vortex-body interactions within the vortex-shedding process. As a result, the combined influence of higher $Re$ and $U^{*}$ contribute to unsteady wake behavior in the tandem cylinder configuration.

The vortex-body interaction process could be further investigated by quantifying the $C_{\mathrm{lv}}$ values in different flow regimes. For the isolated configuration at $Re=60$, the $C_{\mathrm{lv}}$ values are found to be generally positive and close to zero along the entire span of the cantilever, as shown in Fig. 17. The relatively uniform distribution of $C_{\mathrm{lv}}$ indicates a stable hydrodynamic loading in phase with the cantilever’s transverse velocity. In the isolated configuration at $Re=100$, the $C_{\mathrm{lv}}$ values exhibit slightly more variation compared to $Re=60$. While the trend still centers around zero, with the values of $C_{\mathrm{lv}}$ being generally positive, there are more noticeable peaks and troughs along the cantilever’s length. For the tandem configuration at $Re=60$, the $C_{\mathrm{lv}}$ values show a more distinct pattern compared to the isolated cases. There are noticeable peaks and troughs along the length of the cantilever, indicating significant in-phase hydrodynamic excitation. In the tandem configuration at $Re=100$, the $C_{\mathrm{lv}}$ values exhibit the most significant variations among all cases.

The continuous introduction of upstream vortices, i.e., angular momentum, to the flexible cantilever, coupled with the vortex-induced motion of the cantilever, results in complex vortex-structure interactions. This interplay culminates in sustained vibrations of the cantilever, particularly at high reduced velocities within the WIV-dominated regime. With all these findings, we summarize the response amplitude of the cantilever as a function of $Re$ and $U^{*}$ in isolated and tandem configurations in the following subsection.