2.1. Governing equations

The fluid flow is governed by unsteady, incompressible and viscous Navier–Stokes equations, written in non-dimensional form as follows:

(2.1)$$\begin{gather} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} = 0, \end{gather}$$

(2.2)$$\begin{gather}\frac{\partial \boldsymbol{u}}{\partial t} + (\boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{\nabla}) \boldsymbol{u} =-\boldsymbol{\nabla} p + \frac{1}{Re} {\nabla}^2 \boldsymbol{u}, \end{gather}$$

where dimensionless variables include flow velocity $\boldsymbol {u}=\boldsymbol {u}^*/u^*_0$, pressure $p = p^* / \rho _f^* u_0^{* 2}$, time $t = u^*_0 t^* / D^*$ and Reynolds number ($Re = \rho _f^* u^*_0 D^* / \mu ^*$). The superscript $^*$ denotes a dimensional variable, and $u^*_0$, $\rho _f^*$, $D^*$ and $\mu ^*$ are free stream flow velocity, fluid density, cylinder diameter and dynamic viscosity, respectively. Definitions of major symbols are provided in table 1.

Table 1. Definitions of the symbols used in the present study.

We consider the dynamics of two elastically mounted circular cylinders in a tandem arrangement, as shown in figure 1. This set-up is a simplified version of Price & Abdallah (Reference Price and Abdallah1990) for modelling FIV response of electrical conductor wires separated by elastic spacers. Here we consider cross-flow vibrations of cylinders with 1-DOF with equal mass ratio $m$ and zero damping. The equation of motion of an elastically coupled cylinder system is as follows:

(2.3)\begin{equation} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} \ddot{y}_1 \\ \ddot{y}_2 \end{bmatrix} + \begin{bmatrix} k_{1}+k_{2} & -k_{2} \\ -k_{2} & k_{2}+k_{3} \end{bmatrix} \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} = \begin{bmatrix} 4 F_{T 1}/{\rm \pi} m \\ 4 F_{T 2}/{\rm \pi} m \end{bmatrix}, \end{equation}

where $y_i=y_i^*/D^*$ is the non-dimensional transverse amplitude of displacement and $F_{T i} = F_{T i}^*/\rho _f^* u_0^{* 2} D^*L^*$ is the non-dimensional transverse force on the $i$th cylinder. Here we have $i=1$ for the UC and $2$ for the DC. Here $k_{i}=k_i^* D^{* 2}/m_{s}^* u_0^{* 2}$ is the non-dimensional reduced stiffness, and ${4 m_s^*}/{{\rm \pi} \rho _f^* D^{* 2}}L^*$ is the mass ratio ($m_1=m_2=m$) of the cylinders. Here $y_i$, $\dot {y}_i$ and $\ddot {y}_i$ represent the instantaneous displacement, velocity and acceleration of the $i$th cylinder. Here $L^*$ is the cylinder spanwise length and is taken as unity in the present study.

For the conventional tandem cylinder configuration (Case 1), the reduced stiffness values are $k_1=k_3$ and $k_2=0$. Substituting these values in (2.4a,b)–(2.5a,b) gives $f_{n1}=f_{n2}=\sqrt {k_1}/2 {\rm \pi}$, with $\sigma _1=1$ and $\sigma _2=-1$. This implies that the uncoupled tandem cylinder system has a single natural frequency, with any linear combination of the two mode shapes ($[1,1]^{\rm T}$ and $[-1,1]^{\rm T}$) as the associated mode shape. This is why there is no specific mode shape or $\phi _{y12}$ associated with uncoupled tandem cylinders. However, for Cases 2 and 3, elastic coupling is introduced via an additional spring with stiffness $k_2 \neq 0$. Owing to the coupling stiffness $k_2$, the system bifurcates into two natural frequencies ($\,f_{n1}$ and $\,f_{n2}$) and associated modal amplitude ratios ($A_{n1}/A_{n2}=\sigma _{n1}$ and $\sigma _{n2}$, respectively):

(2.4a,b)$$\begin{gather} f_{n 1} = \cfrac{1}{2 {\rm \pi}} \sqrt{ \cfrac{k_1+k_3}{2} + k_2 - \sqrt{ \left( \cfrac{k_1-k_3}{2} \right)^2 + k_2^2}} ,\quad \sigma_{n 1} = \sqrt{1 + \left( \cfrac{k_1-k_3}{2 k_2} \right)^2} - \cfrac{k_1-k_3}{2 k_2}, \end{gather}$$

(2.5a,b)$$\begin{gather}f_{n 2} = \cfrac{1}{2 {\rm \pi}} \sqrt{\cfrac{k_1+k_3}{2} + k_2 + \sqrt{ \left( \cfrac{k_1-k_3}{2} \right)^2 + k_2^2}} ,\quad \sigma_{n 2} =-\sqrt{1 + \left( \cfrac{k_1-k_3}{2 k_2} \right)^2} - \cfrac{k_1-k_3}{2 k_2}. \end{gather}$$

2.2. Simulation set-up

The FIV of two circular cylinders in the tandem configuration is studied in an open domain flow of $Re=100$ as shown in figure 1. The open domain is simulated by enforcing constant inflow velocity on the upstream wall ($u=1,v=0$). The free slip boundary condition ($\partial u/\partial y=0,v=0$) is imposed on the side walls of the considered domain. The right-hand boundary is prescribed with a fully developed boundary condition ($\partial u/\partial x=0,\partial v/ \partial x=0$). Two tandem cylinders, with gap ratio $G$ varying systematically from $1.1$ to $5$, are immersed in the flow. The fluid–structure interface is imposed with no-slip boundary condition ($(\boldsymbol {\nabla } \boldsymbol {u})\boldsymbol{\cdot}\boldsymbol {\hat {n}} = 0$). The cylinders are constricted to move only in the transverse direction. The mass ratio of both cylinders is kept constant at $m=10$, with no damping $\xi =0$.

The reduced stiffness values $k_1,k_2,k_3$ are varied systematically to obtain multiple structural vibration configurations. The excitation force frequency is primarily assumed to be $\approx St_{00} \sim St_0$. Therefore, only the natural frequency closer to $St_0$ is assumed to be excited in a particular case. The stiffness parameters proposed by Ding et al. (Reference Ding, Srinil, Bao, Zhou and Han2020) are used in the present analysis to excite individual modes (see the Appendix) during FIV of tandem cylinders. First, $k_1=k_3\in [0.12,2.47];k_2=0$, corresponding to $U_R = u_0^*/f_{n}^* D^* \in [4,18]$, is used in Case 1. In Case 2, $k_1=6,k_2\in [0.12,8.19];k_3=0$ configuration is used to ensure $f_{n2} \gg St$ for $U_{R}=u_0^*/f_{n1}^* D^* \in [4,18]$. Finally, the Case 3 configuration is realized using $k_1=0.1,k_2\in [0.018,8.19];k_3=0$ such that $f_{n1} \ll St_0$ for $U_{R}=u_0^*/f_{n2}^* D^* \in [4,18]$.

The computational method used in the present study, is briefly described in § S1 of the supplementary material available at https://doi.org/10.1017/jfm.2023.910. Based on the domain and grid independence tests (see § S2 in the supplementary material), a Cartesian grid of $1024 \times 256$ distributed over a domain size of $85 \times 30$, with grid size $\Delta x = \Delta y = 0.02$ in the refined regions, is utilized to perform all the subsequent simulations at time step size $\Delta t = 0.01$. Further, the solver is verified (see § S3 in the supplementary material) and shows good agreement with Ding et al. (Reference Ding, Srinil, Bao, Zhou and Han2020) for elastically coupled FIV of side-by-side cylinders.

2.3. The FIV quantification parameters

Several quantification variables are used to compare and contrast the FIV response across different cases, $G$ and $U_R$. The cylinder displacements $A_i$ correspond to the maximum FIV displacement amplitude, with bars indicating the localized cycle-to-cycle amplitude fluctuations. The maximum coefficient of fluctuating transverse force on the two cylinders is given by

(2.6)\begin{equation} C_{Ti}^\prime =2 F_{Ti}^\prime = 2 (F_{Ti} - \bar{F}_{Ti})_{max}. \end{equation}

The transverse force is composed of potential ($C_{Pi}=-2 C_{a} ({\rm \pi} /4) \ddot {y}_i$) and vortex ($C_{vi} = C_{Ti}-C_{Pi}$) force components (Lighthill Reference Lighthill1986; Govardhan & Williamson Reference Govardhan and Williamson2000). Here $C_{a}$ is the added mass coefficient and is assumed to be ${\approx }1$, corresponding to the isolated cylinder case (Govardhan & Williamson Reference Govardhan and Williamson2000). Further, the stationary $i$th cylinder mean drag $\bar {C}_{Di}$ and maximum fluctuating lift $C_{Li}^\prime$ is also presented for varying $G$.

To remove spurious oscillations in signals of displacement and fluid force, we utilized a Butterworth filter with a non-dimensional cutoff frequency of unity. Typically, $50$ vibration cycles after reaching dynamically steady state were considered for the analysis. The simulations are executed for at least $t = 1200$ and longer, if required, to ensure a dynamic steady state. Some sample signals with delayed dynamic steady state are shown in § S5. The amplitude spectral density (ASD) of displacement ($y_i$) and transverse force ($C_{Ti}$) is normalized using the corresponding structural natural frequency $f = f_{signal}/f_{ni}$ ($i=1$,$1$ and $2$ for Cases 1, 2 and 3, respectively). The ASD of the signals at each $U_R$ is interpolated using histogram distribution on a common frequency scale of $0$ to $4$ with $\Delta f=0.025$ and normalized with the maximum strength of the frequency signal at that $U_R$, and plotted on a logarithmic scale. The colourmap is superimposed with the normalized Strouhal frequency $f_{St_{00}}=St_{00}/f_{ni}$ using a dotted blue line.

Here $\phi _{C_{Ti}}$ and $\phi _{C_{vi}}$ denote phase difference of $C_{Ti}$ and $C_{vi}$ with $y_i$, respectively. Furthermore, $\phi$ is obtained using the phase difference between the ASD frequency component of the two signals, corresponding to the dominant $y_i$ frequency at that $U_R$. Further, the intercylinder phase difference $\phi _{y12}$, $\phi _{C_{T12}}$ and $\phi _{C_{L12}}$ indicate the phase lag of $y_2$, $C_{T2}$ and $C_{L2}$ from $y_1$, $C_{T1}$ and $C_{L1}$, respectively, corresponding to the dominant frequency of $y_1$, $C_{T1}$ and $C_{L1}$. All the phase differences are indicated in degrees.

The angular locations $\theta _i$ are measured from the front stagnation point location of the $i$th cylinder in stationary isolated configuration. The vortex formation length $L_{u^\prime u^\prime }$ is quantified as the downstream distance of the vortex formation point from the centre of the cylinder. The vortex formation point is defined as the location in the flow where the fluctuating component of the axial velocity becomes maximum (Green & Gerrard Reference Green and Gerrard1993). The localized mean coefficient of pressure $C_{Pi}=p^*_i/(1/2 \rho _f^* u_0^{* 2})$ and coefficient of skin friction $C_{Fi}=\tau _i^*/(1/2 \rho _f^* u_0^{* 2})$ is plotted using solid lines, with shaded regions indicating the transient variations in $C_{Pi}$ and $C_{Fi}$ at the corresponding $\theta _i$ locations. The separation (reattachment) $\theta _i$ correspond to a positive to negative (negative to positive) transition of $C_{Fi}$ on the cylinder surface. The localized gap flow velocity $c_g$ is quantified as the flow velocity normal to the line joining the centres of the UC and DC. The transient gap flow $F_g$ is obtained as the total volume flow rate across the line joining UC and the DC. Further, energy transfer from the fluid to the cylinder is quantified as the rate of work done by the flow on the cylinder, $W_i = C_{Ti} \dot {y}_i$.

The dynamic mode decomposition (DMD) is performed on the vorticity field for quasiperiodic FIV responses. We use the methodology and the algorithm described by Schmid (Reference Schmid2010), Sayadi et al. (Reference Sayadi, Schmid, Nichols and Moin2013) and Kutz et al. (Reference Kutz, Brunton, Brunton and Proctor2016). All the field data, corresponding to the mesh columns of the oscillating cylinders, is neglected during DMD analysis and later plotted after linear interpolation for those regions. Here $|\alpha _i|$ shows the strength of corresponding modal frequencies. A typical case consists of at least 500 snapshots for periodic and up to 1800 snapshots for quasiperiodic FIV response at non-dimensional time interval of 0.1. Further details of the methodology have been discussed in Sharma et al. (Reference Sharma, Pandey and Bhardwaj2022b).

3.1. Case 1: elastically uncoupled cylinders

The UC FIV response (figure 2a) for $2 \leq G \leq 5$ is quite similar to the VIV response of an isolated circular cylinder (Prasanth & Mittal Reference Prasanth and Mittal2009). Here $A_1$ shows an initial jump to ${\sim }0.65$ and is gradually reduced to ${\sim }0.2$ during lock-in, with amplitude variations similar to upper and lower branch variations at high $Re$ (Govardhan & Williamson Reference Govardhan and Williamson2000). The initial and final desynchronization have negligible $A_1$ (${\sim }O(10^{-3})$ and ${<}0.05$, respectively). Although some variations in $U_R$ correspond to various jumps in $A_1$, the qualitative features remain consistent for various $G$. While the DC FIV response $A_2$ (figure 2b) exhibits some similarities with the UC, the following are some important differences. The $A_2$ for $G \geq 2$ shows an initial jump, similar to $A_1$. However, $A_2$ is lower (${\sim }0.4$) than $A_1$ (${\sim }0.65$), and continuously increases with $U_R$. Furthermore, $A_2$ shows a sharp jump to ${\sim }1.0$ at $U_R \sim 7$ for all $G \geq 2$, and further gradually decreases with increasing $U_R$. The $A_2$ is much larger (nearly double) than $A_1$ in this regime. Finally, $A_2$ drops to very small amplitudes at large $U_R$, corresponding to the final desynchronization of the UC. However, $A_2$ remains $> 0.1$ in the desynchronization regime for $G > 2$, even at very large $U_R$. The larger $A_2$ is possibly due to the interactions of the UC wake with the DC, resulting in WIV (Assi et al. Reference Assi, Bearman and Meneghini2010, Reference Assi, Bearman, Carmo, Meneghini, Sherwin and Willden2013).

Figure 2. Variation of amplitude with reduced velocity ($U_R$) for (a,c,e) upstream cylinder, $A_1$, and (b,d,f) downstream cylinder, $A_2$: (a,b) Case 1, elastically uncoupled; (c,d) Case 2, elastically coupled, Mode 1; (e,f) Case 3, elastically coupled, Mode 2. Difference cases of gap ratio $G$ are compared in each panel. The insets in (a,c,e) represent the configuration for Case 1, 2 and 3, respectively.

The FIV response for uncoupled tandem cylinders (Case 1) with $2 \leq G \leq 5$ is broadly classified into four regimes. The first regime is initial desynchronization (ID) at very low $U_R$. The second regime corresponds to the amplitude jumps for both cylinders with $A_1>A_2$. The third regime is initiated by a secondary jump in $A_2$, resulting in $A_2>A_1$. Finally, the fourth regime is the final desynchronization of $A_1$ and $A_2 \sim 0.1$, corresponding to WIV. The observed characteristics are consistent with previous numerical (Papaioannou et al. Reference Papaioannou, Yue, Triantafyllou and Karniadakis2008; Borazjani & Sotiropoulos Reference Borazjani and Sotiropoulos2009; Prasanth & Mittal Reference Prasanth and Mittal2009; Bao et al. Reference Bao, Huang, Zhou, Tu and Han2012; Griffith et al. Reference Griffith, Jacono, Sheridan and Leontini2017) and experimental (King & Johns Reference King and Johns1976; Ruscheweyh Reference Ruscheweyh1983; Allen & Henning Reference Allen and Henning2003; Narvaez, Schettini & Silvestrini Reference Narvaez, Schettini and Silvestrini2020; Xu et al. Reference Xu, Wu, Jia and Wang2021) studies. Notably, the transition between the second and third regime occurs at $U_R \sim 7$ for all $G \geq 2$, corresponding to $f_n \sim 0.14$. This is close to $St_{00}$ in the presence of gap vortices (figure S4c) and is consistent with the finding of Prasanth & Mittal (Reference Prasanth and Mittal2009) that the transition occurs when $St_{00} \sim f_n$. It is worth highlighting that Gu et al. (Reference Gu, Xu, Jiang and Yao2023) also studied the effect of $G \in [2,6]$ on 2-DOF vibrations of tandem square cylinders at $Re=100$ and observed qualitatively similar $A_1$ and $A_2$ variation. In their study, the peaks of $A_1 \sim 0.65$ and $A_2 \in [0.85,1]$ are also close to the peaks $A_1 \sim 0.65$ and $A_2 \sim 1.0$ of the present study, even though the cross-sections are significantly different. However, Qin et al. (Reference Qin, Alam and Zhou2019) performed experimental investigations on tandem circular cylinders for systematic variation in $G \in [1.2,6]$ at $Re \sim O(10^4)$. They observed significant changes in $A_1$ and $A_2$ with variation in $G$, which are not reflected in the present study. This is possibly caused due to galloping vibrations occurring at high $Re$, resulting in high FIV amplitudes at high $U_R$. Moreover, they study the damped vibrations ($m \zeta \sim 0.58$), contrary to the present undamped vibration study ($m \zeta = 0$), resulting in very small VIV contributions in their study.

The $G = 1.1$ results significantly differ from the cylinders with larger $G$. The cylinders show an initial jump in $A_1$ and $A_2$ at $U_R = 5.25$. Eventually, both cylinders show continuously increasing FIV amplitudes, with $A_1$ and $A_2$ reaching ${\sim }1.0$ and ${\sim }1.1$ at $U_R = 18$, respectively. A minor jump in both amplitudes is observed at $U_R = 9$. Chung (Reference Chung2017) ($Re = 100$) reported a similar galloping response for independent elastically mounted tandem cylinders corresponding to $G \leq 1.7$. However, some of their characteristics differ from the present study. In Chung (Reference Chung2017), $A_1$ is slightly higher than the present study, and $A_2$ shows a drop at $Ur=10$, which is not observed in the present study. This is likely due to their lower mass ratio ($m=2$) than the present study ($m=10$) and an additional vibration mode of the cylinders in the in-line direction. Similarly, Chen et al. (Reference Chen, Ji, Williams, Xu, Yang and Cui2018) ($Re=100$) observed a galloping response for three tandem cylinders, with diverging amplitudes at $G = 1.2$ and almost constant amplitudes at $G = 1.5$. Kim et al. (Reference Kim, Alam, Sakamoto and Zhou2009) ($Re=O(10^5)$) also reported a similar galloping response for tandem cylinder FIV at $G = 1.1$ and $1.3$, at large $m \zeta$. Interestingly, Qin et al. (Reference Qin, Alam and Zhou2019) observed a continuous galloping response for $G \leq 1.5$ and a delayed galloping response for $G \leq 3$, for high $m \zeta$ tandem cylinders at high $Re \sim O(10^4)$.

3.2. Case 2: elastically coupled cylinders Mode 1

As shown in § 2.1, only Mode 1 (see inset in figure 2c) is excited for Case 2, i.e. the cylinders are bound to vibrate in-phase ($\phi _{y12} = 0$). Figure 2(c) shows a significantly weaker FIV response for the UC, as compared with Case 1. The maximum $A_1 \sim 0.2$ is substantially lower than $A_1 \sim 0.65$, corresponding to Case 1. Here $A_1$ shows an initial jump similar to the initial branch. It is followed by a gradually decreasing $A_1$ similar to lock-in. The drop in $A_1$ at desynchronization is also gradual and not sharp compared with Case 1.

In figure 2(d), the DC attains $A_2 \sim 0.9$ for $G>2$, and slightly lower values for $G = 1.1$ ($A_2 \sim 0.4$) and $G = 2$ ($A_2 \sim 0.6$). The jump in the initial branch for $A_2$ occurs at almost the same $U_R$, corresponding to the transition from regime 1 to 2 in Case 1. The only exception is $G = 2$, for which the jump corresponds to the $U_R$ of the second amplitude jump between regimes 2 and 3 in Case 1. For Case 2, $A_2$ ($G>2$) attains a higher magnitude than their Case 1 counterpart (regime 2). However, the $A_2$ curve, corresponding to the second amplitude jump ($G = 2$), attains a lower magnitude than its Case 1 counterpart (regime 3). The inconsistent behaviour of $G = 2$ is likely due to the unfavourable time delay of vortices between the two cylinders. The $\phi _{C_{L12}}\sim 90^{\circ }$ indicates a slight delay from the desired modal in-phase forcing of Case 2. As the cylinders vibrate, the vortex path between the cylinders elongates, resulting in a more antiphase nature of the fluid forces. This phenomenon is similar to ‘modal decoupling’, reported by Price & Abdallah (Reference Price and Abdallah1990), that leads to FIV suppression even if the excitation frequency is closer to the cylinder's natural frequency. In this context, previous studies on rigidly coupled two tandem cylinders have realized $\phi _{y12}=0$, similar to Case 2. For example, Zhao (Reference Zhao2013) studied the rigidly coupled tandem cylinders at $Re=150$ and observed a similar VIV-type amplitude variation curve. Similar to the present study, the FIV amplitudes in their study ($A_1=A_2$) were smaller for $G = 1.5$ ($A_1 \sim 0.2$) and $2$ ($A_1 \sim 0.6$), as compared with $G \geq 4$ ($A_1 \sim 0.9$). Zhao, Murphy & Kwok (Reference Zhao, Murphy and Kwok2016) also reported a VIV-type response variation in $A_1=A_2$ for rigidly coupled tandem cylinders at $Re=5000$ for $G>1.5$.

While rigid coupling enforces $A_1=A_2$, even if the two cylinders experience different flow forces, we report significant amplitude variations between $A_1$ and $A_2$ for the elastic coupling. This shows the implications of flow force and mode shape interactions between the two cylinders. The $A_1/A_2$ (${\sim }1/3$) variation is introduced in the side-by-side configuration of Ding et al. (Reference Ding, Srinil, Bao, Zhou and Han2020) for the same mode shape. However, $A_1/A_2$ in the present study is ${\sim }0.2$ for tandem configuration with the same structural parameters. Therefore, enhanced transverse forces on the DC modify the relative FIV amplitudes in the present case. A detailed description of flow forces on the cylinders is discussed in § S7.

While $G = 1.1$ (Case 1) shows a galloping-like response (§ 3.1), $G = 1.1$ (Case 2) shows a VIV-like response. Zhao (Reference Zhao2013) also observed this behaviour at $Re=150$ and $G = 1.5$, reporting an exceptionally low FIV amplitude. Galloping is possibly due to $\phi _{y12} \neq 0$ induced between the tandem cylinders in Case 1. The modal arrangement enforces $\phi _{y12} = 0$ in Case 2, possibly suppressing galloping at small $G$. Maeda et al. (Reference Maeda, Kubo, Kato and Fukushima1997) also observed the absence of galloping for rigidly connected tandem cylinders ($m \zeta \sim 0.46$) at $Re \sim O(10^4)$, with a resumption of galloping in a slightly staggered arrangement. However, Zhao et al. (Reference Zhao, Murphy and Kwok2016) observed a galloping-like response for the rigidly connected tandem cylinders ($m = 2.5, \zeta = 0$) at $G = 1.5$ in an $Re=5000$ flow. Similar behaviour is also reflected in the FIV response of elliptical cylinders (single body approximation for closely placed tandem cylinders), as a narrow lock-in at low $Re$ (Yogeswaran et al. Reference Yogeswaran, Sen and Mittal2014) and an elongated lock-in at high $Re$ (Zhao, Hourigan & Thompson Reference Zhao, Hourigan and Thompson2019). Therefore, $\phi _{y12}=0$ configuration tandem cylinders at very small $G$ can exhibit galloping response at sufficiently smaller $m$ and larger $Re$.

3.3. Case 3: elastically coupled cylinders Mode 2

Similar to Case 2, only Mode 2 (see inset in figure 2e) is excited for Case 3, i.e. the cylinders are bound to move out-of-phase ($\phi _{y12} = 180^{\circ }$). The UC with $G \geq 2$ attains $A_1 \sim 0.9$ during the lock-in, which is larger than the Case 1 or 2 counterparts. Similarly, the DC attains $A_2 \sim 0.75$ for $G>2$ in the lock-in regime. Similar to Case 2, both $A_1$ and $A_2$ do not show any amplitude jump in the middle of the lock-in branch, contrary to Case 1. The initiation of the lock-in is reflected by a sharp jump in $A_1$ and $A_2$ for $G = 2$, which is synchronized with the first amplitude jump in Case 1. The jump in $A_1$ and $A_2$ during the lock-in initiation becomes increasingly gradual for $2 \leq G \leq 5$. Interestingly, the reduction gradient of $A_1$ and $A_2$ towards the end of lock-in becomes steeper for $2 \leq G \leq 5$.

Like Cases 1 and 2, the DC exhibits WIV ($A_2 \sim 0.1$) in the high $U_R$ desynchronized regime for $G \geq 3$ in Case 3. Surprisingly, for this Case, the lock-in regime for $G = 2$ extends up to $U_R = 13$, beyond its Case 2 counterpart. This is slightly beyond $U_R = 11.5$, corresponding to the desynchronization of $G = 2$ tandem cylinders in Case 1. Furthermore, an initial local maxima of $A_1 \sim A_2 \sim 0.5$ is attained in the low $U_R$ regime for $G \geq 4$, with significant amplitude fluctuations (§ 4.3). This is similar to the initial branch behaviour in VIV. However, instead of growing monotonically, $A_1$ and $A_2$ reach a local maxima and decrease until lock-in begins.

Furthermore, $\phi _{y_{12}}= 180^{\circ }$ has been reported for connected tandem cylinders. Examples include controlling relative motion using complex thread linkages (Brika & Laneville Reference Brika and Laneville1997; Laneville & Brika Reference Laneville and Brika1999) or pivoting the rigidly coupled cylinders at the centre (Arionfard & Nishi Reference Arionfard and Nishi2018b,Reference Arionfard and Nishia). Laneville & Brika (Reference Laneville and Brika1999) observed multiple peaks for the FIV of cylinders at $Re \sim O(10^4)$ and $G>10$, with significant hysteresis in the low $U_R$ peaks. Similarly, Arionfard & Nishi (Reference Arionfard and Nishi2018b) observed two vibrations amplitude peaks at $U_R = 5.5$ and $7.3$ for $G = 3.9$ at $Re \sim O(10^3)$. While these studies are not exactly similar to the present case, they provide a reasonable correlation with the observations of Case 3.

Interestingly, the $G = 1.1$ in Case 3 shows a galloping response similar to Case 1. The amplitude jumps in $A_1$ and $A_2$ occur at the same $U_R$ compared with the Case 1 counterpart. While the FIV amplitudes in Case 3 are lower than in Case 1, the relative amplitudes between the two cylinders ($A_1/A_2 \sim 0.7$) are very similar. The kinks in $A_i$ observed at $U_R \geq 13$ correspond to the amplitude fluctuations (§ 4.4).

4.1. The FIV regimes of Case 1

As observed in § 3, the overall FIV response in Case 1 ($G > 1.1$) is classified broadly into the following regimes: IDG and ID; UB and LB lock-in; FD or WIV. Since FIV responses for the gap ratios ($G > 1.1$) of Case 1 are qualitatively similar, we discuss a representative case of $G = 4$. Figures 4(a) and 4(b) show that $A_1$ and $A_2$ are negligible (${<}0.01$) in the ID regime ($U_R \in [4,5.25]$). The $C_{T1}^\prime$ and $C_{V1}^\prime$ remain negligible (${\sim }0.01$) with increasing $U_R$ in this regime, with $C_{T2}^\prime$ and $C_{V2}^\prime$ having finite decreasing magnitude. Interestingly, these flow forces are significantly lower than the stationary cylinder counterpart ($C_{L_0}^\prime \sim 0.403$) at $G = 4$, indicating stronger FIV suppression characteristics with increasing $U_R$. The $f_{y1}$, $f_{y2}$, $f_{C_{T1}}$ and $f_{C_{T2}}$ indicate single dominant frequency component parallel to $f_{St_{00}}$, with a slightly lower frequency. The lower frequency results from the absence of gap vortices in this regime (shown in § 6), contrary to the stationary cylinder flow with gap vortices. Finally, $\phi _{C_{Ti}}$ and $\phi _{C_{Vi}}$ remain ${\sim }0^{\circ }$ in this regime. Therefore, this regime is identified as the ID regime.

Figure 4. The FIV characterization for Case 1 at $G = 4$: (a,b) shows $A_1$ and $A_2$, with bars representing cycle-to-cycle amplitude variation. Variation of $C_{Ti}^\prime$ (dotted blue), $C_{Vi}^\prime$ (dotted red) and $C_{Li}^\prime$ (dotted black) are plotted for the (c) upstream ($i=1$, a,c,e,g,i) and (d) downstream ($i=2$, b,d,f,h,j) cylinders. Contours show $f_{yi}$ (e,f) and $f_{C_{Ti}}$ (g,h) with the dotted blue line representing $f_{St_{00}}$ for $G = 4$. Here $\phi _{C_{Vi}}$ (dotted red) and $\phi _{C_{Ti}}$ (dotted blue) are plotted for (i) upstream and (j) downstream cylinders.

The second regime ($U_R \in [5.5,6.75]$) shows sharp jumps in $A_1$ and $A_2$ during the transition from ID regime to this regime, indicating a narrow IB regime near $U_R =5.5$. The amplitudes $A_1 \sim 0.6$ (nearly constant) and $A_2 \sim 0.4$ (gradually increasing) remain high throughout this regime, with $A_1>A_2$. After a sudden jump in flow forces, both the cylinders show a decreasing $C_{Ti}^\prime$ and an increasing $C_{Vi}^\prime$. Dominant frequency components of $f_{yi}$ and $f_{C_{Ti}}$ are observed at ${\sim }1$ in this regime, with additional third harmonic signatures in $f_{C_{Ti}}$. Furthermore, $\phi _{C_{Vi}}$ jumps from $0^{\circ }$ to $180^{\circ }$ at the start of this regime, with $\phi _{C_{Ti}} \sim 0^{\circ }$ throughout this regime. Therefore, this regime is identified as the UB regime.

The $A_1$ starts gradually decreasing in the third regime ($U_R \in [7,8.75)$). The $A_2$ jumps to ${\sim }1.0$ during the transition and remains nearly constant in this regime. After reaching a local maxima, $C_{V1}^\prime$ gradually reduces, with much smaller $C_{T1}^\prime$. The $C_{V2}^\prime$ also shows a gradual reduction in this regime after a sharp jump during the transition from the UB regime. The $f_{yi}$ and $f_{C_{Ti}}$ show dominant components at ${\sim }1$, with third harmonic signatures in $f_{C_{Ti}}$. Additionally, $\phi _{C_{Ti}}$ transitions from $0^{\circ }$ to $180^{\circ }$ in this regime, with $\phi _{C_{Vi}} \sim 180^{\circ }$. Therefore, this regime is identified as the LB lock-in regime. While in Case 1, $A_2$ is significantly larger in LB than UB, we choose to designate it as LB for continuity.

The initiation of the fourth regime ($U_R > 8.75$) is characterized by a sudden initial drop in $A_1$ (${<}0.07$) and $A_2$ (${<}0.27$), followed by a gradual reduction with increasing $U_R$. The $C_{T1}^\prime$ and $C_{T2}^\prime$ asymptotically approach $C_{L1}^\prime$ and $C_{L2}^\prime$, respectively. The $f_{yi}$ and $f_{C_{Ti}}$ shows single frequency components coinciding to $f_{St_{00}}$, with $\phi _{C_{Ti}}$ and $\phi _{C_{Vi}}$ at $180^{\circ }$. The spectral characteristics show desynchronization, with $A_2 > A_1$ and $C_{T2}^\prime \gg C_{T1}^\prime$. This indicates that the DC experiences an additional FIV excitation due to the wake of the UC. Therefore, this regime is identified as the WIV regime, continuing with the nomenclature proposed by Assi et al. (Reference Assi, Bearman, Carmo, Meneghini, Sherwin and Willden2013).

4.2. The FIV regimes of Case 2

The ID regime of Case 2 is characterized by negligible $A_1$ and $A_2$ ($< 0.01$) for $U_R \in [4,5.5]$. The $C_{T1}^\prime$, $C_{V1}^\prime$ are of $O (10^{-3})$, and $C_{T2}^\prime$, $C_{V2}^\prime$ of $O(10^{-1})$; with decreasing magnitude for increasing $U_R$. Dominant components of $f_{yi}$ and $f_{C_{Ti}}$ are parallel to $f_{St_{00}}$ with slightly lower frequency due to the absence of gap vortices. Similar to Case 1, Case 2 exhibits FIV suppression characteristics in this regime. Interestingly, $\phi _{C_{T1}} \approx \phi _{C_{V1}} \sim 80^{\circ }$ and $\phi _{C_{T2}} \approx \phi _{C_{V2}} \sim 360^{\circ }$ shows a deviation from the expected $0^{\circ }$ phase difference response in this regime.

The $U_R \in [5.5, 6.0]$ shows a gradual increase in $A_i$, accompanied by sharp increase in $C_{Ti}^\prime$ and $C_{Vi}^\prime$. The $A_i$ show minor cycle-to-cycle fluctuations (indicated by bars in figure 5a,b), which are reflected in the ASD characteristics $f_{yi}$ and $f_{C_{Ti}}$. Therefore, this regime corresponds to the IB.

Figure 5. The FIV characterization for Case 2 at $G = 4$. The rest of the caption is the same as in figure 4.

Similar to the isolated cylinder lock-in branch, $U_R \in [6.0,7.75]$ shows steady and high values of $A_1$ (${\leq }0.2$) and $A_2$ (${\leq }1.0$). The ASD characteristics show dominant $f_{yi}$ and $f_{C_{Ti}}$ close to $1$, with third harmonic signatures in $f_{C_{Ti}}$. However, the ASD signatures show a minor gradient in associated frequencies with $U_R$. Further, the magnitude ($C_{Ti}^\prime$ and $C_{vi}^\prime$) and phase ($\phi _{C_{Ti}}$ and $\phi _{C_{vi}}$) of flow forces do not exhibit any explicit UB- or LB-like behaviour. For example, the decreasing $C_{T1}^\prime$ and $C_{V1}^\prime$ indicate a lower branch behaviour, however, $C_{T2}^\prime$ and $C_{V2}^\prime$ variations do not confirm to that behaviour. Furthermore, $\phi _{C_{T}}$ and $\phi _{C_{v}}$ of a freely oscillating cylinder (e.g. isolated cylinder or Case 1 configuration) vary between $0^{\circ }$ and $180^{\circ }$. The phase of forces in Cases 2 and 3 may vary anywhere between $0^{\circ }$ and $360^{\circ }$ due to the presence of elastic coupling between the cylinders. In this context, Zhao (Reference Zhao2013) also observed multiple phase jumps for the rigidly coupled cylinders. Therefore, the conventional classification of branches via flow force magnitude and phase is invalid for Cases 2 and 3. We refer to this regime as LN.

The $U_R > 8$ regime of Case 2 shows a WIV response, similar to Case 1. Here $A_1 < 0.01$ and $A_2 \sim 0.13$ are observed at high $U_R$, with $C_{T1}^\prime$ and $C_{T2}^\prime$ asymptotically approaching $C_{L1}^\prime$ and $C_{L2}^\prime$, respectively. The $f_{yi}$ and $f_{C_{Ti}}$ show dominant frequencies along $f_{St_{00}}$, indicating a WIV regime. The $\phi _{C_{T2}} \approx \phi _{C_{V2}}$ remain $180^{\circ }$ for the whole regime. However, $\phi _{C_{T1}} \approx \phi _{C_{V1}}$ initially remains $180^{\circ }$ up to $U_R = 14$ and reduces to $0^{\circ }$ beyond $U_R = 15$.

4.3. The FIV regimes of Case 3

The $G \leq 3$ in Case 3 shows the ID regime similar to Cases 1 and 2. However, if $G$ is increased beyond $3$, a local maxima of $A_1$ and $A_2$ is observed in the low $U_R$ regime, as shown in figure 2. This regime ($U_R < 5.75$) for $G = 4$ in Case 3 is characterized by large $A_1$ (up to $0.6$) and $A_2$ (up to $0.55$), accompanied by large cycle-to-cycle amplitude fluctuations as well (figure 6). In this regime, unlike IB, amplitudes increase, attain a local peak and then decrease towards the end of the regime. The $C_{Ti}^\prime$ and $C_{Vi}^\prime$ also attain large magnitudes, indicating loss of FIV suppression characteristics at low $U_R$ for Case 3. The ASD signatures of $f_{yi}$ and $f_{C_{Ti}}$ indicate multifrequency signatures, similar to the IB VIV response. However, plotting the in-phase modal frequency (dotted black line) corresponding to Case 2 indicates that the multiple frequencies present in the system are the two structural natural frequencies ($f_{n1}$ and $f_{n2}$) and one Strouhal frequency ($St_{00}$). Therefore, the initial local peak observed for $G \geq 4$ in Case 3 is caused by the excitation of both the structural modes at very low $U_R$. The in-phase mode gets excited at low $U_R$ because of the close proximity of $f_{n1}$ to $St_{00}$ at very low $U_R$ (see the Appendix). Therefore, this regime is referred to as a MM regime. The MM is only observed for sufficiently large gap ratios ($G \geq 4$) in Case 3, and its excitation is possibly associated with the gap vortex formation.

Figure 6. The FIV characterization for Case 3 at $G = 4$. A dotted black line in (e,g,f,h) shows normalized first mode (in-phase) frequency. The rest of the caption is the same as in figure 4.

The second regime ($U_R \in [5.75,13)$) is characterized by a gradual increase in $A_1$ and $A_2$ from ${\sim }0.16$ and ${\sim }0.03$, respectively. The $A_1$ and $A_2$ increase up to ${\sim }0.90$ and ${\sim }0.73$, respectively, and then gradually decrease until $U_R = 13$, with minor amplitude fluctuations. Similar to Case 2, no general pattern is observed in the variation of $C_{Ti}^\prime$ and $C_{Vi}^\prime$. Dominant components of $f_{yi}$ and $f_{C_{Ti}}$ are observed at $1$, with significant third harmonic signatures in $f_{C_{Ti}}$. Surprisingly, even harmonics are also present in $f_{C_{Ti}}$, with stronger signatures in the large $A_i$ regions. However, we cannot conclusively predict the origin of these even harmonics in the present study. As explained in Case 2, the force and phase signatures are significantly modified due to elastic coupling between the cylinders and are not useful in inferring branching characteristics. Therefore, this regime is referred to as the LN regime.

Like Cases 1 and 2, the $U_R \geq 13$ regime is called the WIV regime. The $A_2$ (${\sim }0.1$) is significantly larger than $A_1$ (${\sim }0.025$) in this regime, with $C_{T1}^\prime$ and $C_{T2}^\prime$ approaching $C_{L1}^\prime$ and $C_{L2}^\prime$, respectively. The $f_{yi}$ and $f_{C_{Ti}}$ show desynchronized FIV response, with dominant frequency component along $f_{St_{00}}$. The $\phi _{C_{Ti}}$ and $\phi _{C_{Vi}}$ remain close to $180^{\circ }$ throughout this regime.

4.4. Galloping at $G = 1.1$

As discussed in § 3, tandem cylinders with $G = 1.1$ exhibit galloping in Cases 1 and 3 (figure 7). While the amplitude responses for both cases are qualitatively similar, some notable differences are discussed as follows. The first regimes ($U_R < 5.25$ in Case 1 and $U_R < 5.75$ in Case 3) are quite similar to the ID regime of the FIV systems discussed in §§ 4.1, 4.2 and 4.3. The $A_1 \sim A_2 \sim O (10^{-3})$ and $C_{T1}^\prime \approx C_{V1}^\prime \sim 0.03$, with continuously decreasing $C_{T2}^\prime \approx C_{V2}^\prime$ for increasing $U_R$, indicates the presence of FIV suppression characteristics. The $f_{yi}$ and $f_{C_{Ti}}$ have dominant frequency components along $f_{St_{00}}$, without any offset. Further, Case 1 is characterized by $\phi _{y1} \sim \phi _{y2} \sim \phi _{C_{T1}} \sim \phi _{C_{T2}} \sim \phi _{C_{V1}} \sim \phi _{C_{V2}} \sim 0$ in this regime. The phase characteristics of Case 3 are not considered due to the elastic coupling effects discussed in §§ 4.2 and 4.3.

Figure 7. The FIV characterization at $G = 1.1$ for Case 1 and 3; (i,ii) and (iii,iv) plot the data of these Cases, respectively. The rest of the caption is the same as in figure 4.

The second regime, corresponding to $U_R \in [5.25,9.0]$ of Case 1, shows LN characteristics, similar to $G \geq 2$ of Case 1 (combined UB and LB). The $A_1$ and $A_2$ jump at the start of this regime and maintain a high amplitude throughout the regime. After an initial jump, $C_{Ti}^\prime$ and $C_{Vi}^\prime$ show gradual variations similar to Case 1 $G \geq 2$. The $f_{yi}$ and $f_{C_{Ti}}$ also show dominant frequency components along $1$, with third harmonic components in $f_{C_{Ti}}$. The $\phi _{C_{Ti}}$ and $\phi _{C_{Vi}}$ of Case 1, along with $C_{Ti}^\prime$ and $C_{Vi}^\prime$ variations, indicate separate UB and LB behaviour. However, the LN regime for Case 1 $G = 1.1$ is not divided into UB and LB regimes due to no significant change in $A_i$ (observable for Case 1 $G \geq 2$ in § 4.1).

The second regime of Case 3 ($U_R \in [5.75,12]$) also shows an initial jump, followed by gradually increasing $A_1$ and $A_2$. The $f_{y1}$, $f_{y2}$ and $f_{C_{T1}}$, $f_{C_{T2}}$ also show dominant frequency components along $1$, with third harmonic components in $f_{C_{Ti}}$ similar to Case 1 LN regime. However, due to the elastic coupling, the flow force and phase signals are modified (§ 4.3) and cannot be directly used for further regime bifurcation. Therefore, we consider this regime Case 3 LN for $G = 1.1$.

The third regime ($U_R > 9$ in Case 1 and $U_R > 12$ in Case 3) is characterized by a steady and increasing $A_i$. Case 3 shows significant cycle-to-cycle amplitude fluctuations, similar to those observed for the D-section cylinder galloping vibrations (Sharma et al. Reference Sharma, Garg and Bhardwaj2022a). The $C_{Ti}^\prime$ is steady for Case 1 (similar to FD or WIV regime) and shows irregular variations in $C_{Ti}^\prime$ with $U_R$ for Case 3 (similar to single cylinder galloping). For Case 1, dominant components of $f_{yi} \approx f_{C_{Ti}}$, are along 1, with third harmonic signatures in $C_{Ti}$. Interestingly, $f_{yi}$ and $f_{C_{Ti}}$ of Case 3 show frequency components parallel to $f_{St_{00}}$, similar to the galloping ASD characteristics of a D-section cylinder (Sharma et al. Reference Sharma, Garg and Bhardwaj2022a). Therefore, this regime is called GD. Although ASD characteristics of Case 1 do not show desynchronization characteristics, steady $C_{Ti} \sim 0.5$ indicates a significantly lower dominance of VIV in this regime. Therefore, this regime is considered GD for Case 1 as well.

6.1. Desynchronization and WIV regimes

The ID regime is discussed for $G = 5$ of Case 1 at $U_R = 4$. Figure 9(ai) shows negligible $y_1$ ($A_1 \sim O(10^{-4})$) and $y_2$ ($A_2 \sim 0.004$) in the ID regime, along with much smaller transverse force ($C_{T1}^\prime \sim 0.006$ and $C_{T2}^\prime \sim 0.12$) compared with the stationary tandem $G = 5$ configuration ($C_{L1}^\prime \sim 0.38$ and $C_{L2}^\prime \sim 1.52$). The vortex shedding patterns (figures 9aiv–9avi) show the absence of gap vortices at $G = 5$, larger than the critical value of $G_c=4$. The vortex formation point (shown by diamond symbols in the downstream wake) is $L_{u'u' 2} \sim 5.6$ from the centre of the DC, which is larger than $L_{u'u'} \sim 3.0$, corresponding to a stationary single cylinder (Green & Gerrard Reference Green and Gerrard1993). These observations further confirm the dominance of FIV suppression in this regime. Figure 9(aii) indicates a flow separation for the UC at $\theta _{1} \sim 114^{\circ }$, with stationary rear stagnation point. This can be correlated with minor fluctuations in $C_{F1}$. While $C_{P1} \sim 1.1$ at the front stagnation point and flow separation occurring at $\theta _{1} \sim 114^{\circ }$ corresponds to the isolated stationary cylinder configuration, $C_{P1} \sim -0.4$ at the rear stagnation point is slightly larger. Interestingly, the variation of $C_{P1}$ and $C_{F1}$ resemble the findings of Chopra & Mittal (Reference Chopra and Mittal2019) for a ‘steady flow’ (no vortex shedding) across an isolated circular cylinder at $Re=100$. This indicates that the changes in the rear stagnation region of the UC are only due to the absence of gap vortices. The DC exhibits $C_{P2} < 0$ (figure 9aiii), indicating total submergence of the DC in the shear layers emanating from the UC. The shear layers from the UC reattach onto the DC at $\theta _{2} \sim 51^{\circ } \pm 4^{\circ }$. Similarly, the DC front stagnation point also shows fluctuations ranging in $\theta _{2} \sim 0^{\circ } \pm 10^{\circ }$. While the front stagnation and reattachment points show fluctuations on the DC, we do not observe intermittent shedding in the gap for the ID regime in the present study. The flow remains attached to the DC up to $\theta _{2} \sim 129^{\circ }$, with transient reattachment and separation occurring on the downstream surface based on the vortex shedding timing. $C_{P2}$ and $C_{F2}$ show significant transient fluctuations (min/max indicated by shaded regions) on the DC (especially around $\theta _2 \sim 90^{\circ }$), resulting in significantly larger $C_{T2}$ than $C_{T1}$.

The IDG regime at $U_R = 4$ for $G = 5$ is shown for Case 2 in figures 9(bi)–9(bvi). Both $y_1$ and $y_2$ remain small ($A_1 \sim 0.028$ and $A_2 \sim 0.045$, respectively) during IDG, even in the presence of gap vortices (figures 9biv–9bvi). However, $y_1$ and $y_2$, in this case, are significantly larger than the ID counterparts (figure 9ai), discussed earlier.

As discussed in §§ 4.2 and 4.3, $\phi _{C_{T1}}$ and $\phi _{C_{T2}}$ get significantly modified from $0^{\circ }$ due to the presence of elastic coupling between the cylinders. The flow separation occurs on the UC at $\theta _{1} \sim 118^{\circ } \pm 10^{\circ }$, with large fluctuations in $C_{F1}$ and $C_{P1}$ on the leeward side of the cylinder. Further, $C_{P1} \sim -0.64 \pm 0.05$ at $\theta _1 \sim 180^{\circ }$ is of the same order as $\bar {C}_P$ of the isolated stationary cylinder at $Re=100$ (Chopra & Mittal Reference Chopra and Mittal2019). The DC encounters large fluctuations in $C_{P2}$ and $C_{F2}$, with narrow steady flow reattachment in $\theta _2 \in [45^{\circ }, 90^{\circ }]$. Further, $C_{P2}$ varies from $-1.3$ to $0.8$ due to intermittent submergence of the cylinder in the upstream wake, indicating the flow of vortices across the DC. Figures 9(biv)–9(bvi) also show the formation of vortices in the gap, which later flow over the DC. However, the vortices are formed at $L_{u'u'1} \sim 1.8$ for the first cylinder, which is much smaller than the isolated stationary cylinder counterpart. Further, the vortex formation point corresponding to the DC ($L_{u'u'2} \sim 0.4$) is very close to the cylinder surface. This indicates that the DC shear layers rapidly merge with the upstream vortices and do not significantly modify the upstream vortex generation/propagation.

The FD regime for Case 1 ($G = 2$) at $U_R = 18$ (figures 9ci–9cvi) is characterized by small $y_1$ ($A_1 \sim 0.0015$) and $y_2$ ($A_2 \sim 0.005$), along with small $C_{T1}$ ($C_{T1}^\prime \sim 0.009$) and $C_{T2}$ ($C_{T2}^\prime \sim 0.03$). The $C_{P1}$ and $C_{F1}$ indicate steady flow across the UC, with stationary front and rear stagnation points. Like an isolated stationary cylinder, $C_{P1}$ at the front stagnation point is ${\sim }1.1$, and flow separation occurs at $\theta _1 \sim 114^{\circ }$. However, $C_{P1}$ at the rear stagnation point drops to $\theta _1 \sim -0.5$ in the absence of gap vortices, which is slightly lower than isolated stationary cylinder under ‘steady flow’ at $Re=100$ (Chopra & Mittal Reference Chopra and Mittal2019). The $C_{P2} < 0$ on the cylinder surface indicates complete submergence of the DC in the UC shear layers. The flow reattaches and separates on the DC at $\theta _2 \sim 61^{\circ }$ and $\theta _2 \sim 130^{\circ }$, respectively. Minor transient variations are observed in $C_{P2}$ and $C_{F2}$ on the surface. As shown in figures 9(civ)–9(cvi), the gap vortices are absent, and the shear layers are directly advected past the DC. Finally, the vortex formation occurs at $L_{u'u'2} \sim 10.2$.

The FD regime in the presence of gap vortices is transformed into the WIV regime, and its characteristics are plotted for Case 1 at $G = 5$, $U_R = 18$ in figures 9(di)–9(dvi). The $y_2$ ($A_2 \sim 0.12$) is significantly larger than $y_1$ ($A_1 \sim 0.03$), with $C_{T2}$ ($C_{T2}^\prime \sim 1.35$) being significantly larger than $C_{T1}$ ($C_{T1}^\prime \sim 0.32$). The flow forces in the WIV regime are similar to the stationary tandem cylinder counterparts ($C_{L1}^\prime \sim 0.38$ and $C_{L2}^\prime \sim 1.52$ (see figure S5)) with slightly lower magnitudes in the former. The UC shows $C_{P1}$ and $C_{F1}$ variations similar to the isolated stationary cylinder; with $C_{P1} \sim 1.1$ at front stagnation point, $C_{P1} \sim 0.56$ at rear stagnation point, and flow separation at $\theta _1 \sim 117^{\circ }$. Similar to the IDG regime, the DC in the WIV regime shows significant fluctuations in $C_{P2}$ and $C_{F2}$ ranging in $[-1.4,0.84]$ and $[-0.36,0.36]$, respectively; with no steady flow reattachment. This shows that the DC is immersed in the wake of the UC, as shown in figures 9(div)–9(dvi). The vortex formation occurs at $L_{u'u'1} \sim 2.1$ and $L_{u'u'2} \sim 0.6$ for the upstream and downstream cylinders, respectively. Here $L_{u\prime u \prime 1}$ is slightly larger for the WIV regime, as compared with the IDG regime at the same $G = 5$, with both being smaller than the isolated stationary cylinder case. Thus, motion of the DC influences gap vortex formation more than that by the UC.

6.2. Lock-in regimes

6.2.1. The UB regime: Case 1

The UB lock-in is discussed for $G = 3$, $U_R = 6$ for Case 1 as follows. As shown in figure 10(a), $y_1$ ($A_1 \sim 0.62$) is larger than $y_2$ ($A_2 \sim 0.39$). Similarly, $C_{T1}^\prime \sim 0.61$ is also larger than $C_{T2}^\prime \sim 0.39$. Figures 10(c) and 10(d) indicate the presence of strong odd harmonics in the signals of $C_{T1}$ and $C_{T2}$, which results in a slight distortion of the transverse force signals. The downstream wake is characterized by a $C(2S)$ vortex shedding pattern (figure 10e).

Figure 10. Transient FIV characteristics of the Case 1 UB regime. (a) Time variation of $y_1$, $y_2$, $C_{T1}$ and $C_{T2}$. (b) Transient variation of $W_1$, $W_2$ and $F_g$. (c) The ASD (without normalization) for $y_1$ and $C_{T1}$. (d) The ASD (without normalization) for $y_2$ and $C_{T2}$. (e) Overall vortex shedding pattern. (f) Transient flow characteristics. The upper subpanels show vorticity, with arrows representing gap flow velocity $c_g$. The orange and green coloured boundaries represent anticlockwise (ACW) and clockwise (CW) shear force on the cylinder surface. The lower subpanels show pressure distribution, and the arrows represent surface pressure distribution on the cylinders. See also supplementary movies of transient wake animations for the UB regime of Case 1 (movie5.mp4).

Figure 10(b) shows that energy is dissipated by the UC ($W_1<0$) during dominant $F_g$ for $G=3$. This behaviour is consistent across all cases of $G$ (see figure S7). This implies that the UC FIV is responsible for creating the gap flow. Further, the transient flow snapshots in figure 10(f) show that the gap flow advects downstream in the form of a wave, and is discussed as follows.

For $t_0$, when the UC is at the topmost position, gap flow velocity ($c_g$), close to the UC, decays. The upwards flow from the previous half-cycle leads to delayed (early) flow separation on the lower (upper) side of the UC. This leads to a low-pressure region formation on the upper side of the UC. Simultaneously, the DC receives a downward flow from the UC, and energizes (weakens) the flow on the lower (upper) side of the DC. This causes an early separation of flow from the upper side and flow reattachment on the lower side, resulting in a pressure drop on the upper side of the DC.

For $t_1$ to $t_2$, as the UC moves downward, the rapid change from upward to strong downward $c_g$ results in strong shear layer formations and a sharp drop in pressure on the rear lower side of the UC. UC dissipates energy to the flow ($W_1<0$) during this interval, showing that the UC motion induces gap flow (not the other way round). Simultaneously, the DC continuously receives downwards $c_g$ with reducing magnitude, caused by the approaching CW vortex from the upstream. This pushes the shear layer towards the ACW vortex below the DC. This creates a low-pressure region on the lower rear side of DC, resulting in a mutual pulling force between the ACW vortex and the DC. It should be noted that the flow on the lower side of the DC remains attached due to large $\dot {y}_2$ downwards.

For $t_2$ to $t_4$, $\dot {y}_1$ starts reducing after the UC crosses the mean position. However, the large upwards $c_g$ leads to the pulling of the shear layer from the lower side of the UC, resulting in earlier flow separation and further reduction in pressure at the lower rear side of the UC. As this pressure drop supports the motion of the cylinder, the UC regains the energy from the flow ($W_1>0$) to sustain its FIV. The DC receives a gradually increasing upwards $c_g$ during this interval of downwards motion, marked by an elongated CW upstream vortex. As the DC moves downwards, it suppresses $c_g$ and breaks the CW vortex into a larger part moving upwards and a smaller part moving downwards. This causes a weakening of the lower shear layer, resulting in a downward shifting of the front stagnation point and early flow separation on the lower side of the DC. Simultaneously, surface pressure variation shows that the ACW vortex on the lower side keeps pulling the DC downwards, as it advects downstream. The gain in energy by the DC from the flow ($W_2>0$) during this interval shows that the primary cause for the sustained FIV of the DC is the surface pressure variations caused by the upstream vortices. Further, the early separation of the flow on the lower side of the DC allows reattachment of the upper shear layer, resulting in a mirror image of $t_0$ configuration.

Overall, the UC response is similar to that of an isolated cylinder (Bourguet & Jacono Reference Bourguet and Jacono2014; Chen et al. Reference Chen, Ji, Alam, Xu, An, Tong and Zhao2022), in which (in general), the cylinder dissipates energy to the flow while moving towards the mean position, and regains energy from the flow while moving away from the mean position. Whereas, the DC sustains its FIV amplitude by interacting with the upstream vortices, while the vortices are advected directly to the downstream of the DC.

6.2.2. The LB regime: Case 1

The LB lock-in is discussed for $G = 3$ of Case 1 at $U_R = 7.5$ as follows. Figure 11(a) shows $y_2$ ($A_2 \sim 1.1$) significantly larger than $y_1$ ($A_1 \sim 0.44$), with a similar trend followed by $C_{T2}^\prime \sim 0.5$ being much larger than $C_{T1}^\prime \sim 0.11$. As observed in figure 10, $C_{T1}$ and $C_{T2}$ show minor third-harmonic signatures in the UB regime. On the contrary, figure 11(c,d) show distinct dominance of the third harmonic in the $C_{T1}$ and $C_{T2}$ signals for the LB regime. The overall downstream vortex shedding pattern is also $C(2S)$ (figure 11e), with a lower degree of coalescence of vortices than the UB regime. Further, LB shows a strong dependence on the gap flow characteristics between the tandem cylinders, on the resulting FIV response. The different instances are plotted in figure 11(f) and are described as follows.

Figure 11. Transient FIV characteristics of the Case 1 LB regime. The rest of the caption is same as in figure 10. See also supplementary movies of transient wake animations for the LB regime of Case 1 (movie6.mp4).

For $t_0$, when the UC and the DC are at the top and bottom extremes, respectively, the gap flow becomes maximum due to the staggered cylinder positions. The upper shear layer of the UC gets pushed away from the gap between the cylinders due to the gap flow, and starts to reattach to the top surface of the DC. Simultaneously, the gap flow pushes the lower shear layer into the gap between the cylinders, causing flow separation on the lower side of the UC in ($t_0, t_1$). As a result, a low-pressure region is formed on the lower side of the UC resulting in a downward pull on the cylinder.

For $t_1$, as the UC moves downwards, the gap flow gradually pushes the low-pressure core of lower shear layer from the lower side to the rear side, causing a net gain in energy of the UC from the flow ($W_1>0$) in ($t_0, t_1$). However, as $\dot {y}_1$ increases, the front stagnation point of the UC shifts downwards, resulting in the dissipation of energy ($W_1<0$) for ($t_1,t_2$). Concurrently, the upper shear layer from the UC attaches to the rear side of the upper shear layer of DC, forming a strong low-pressure region on the top rear surface of the DC. As a result, the cylinder gets rapidly pulled upwards, and a net energy gain for the DC ($W_2>0$) occurs in ($t_1, t_2$).

For $t_2$, as the UC crosses the mean position, $\dot {y}_1$ starts reducing, causing the front stagnation point to shift upwards. Since the lower shear layer is still weakly attached to the UC, it slightly energizes the cylinder ($W_1>0$) in ($t_2, t_3$). At the same time, due to large $\dot {y}_1$ in ($t_0, t_2$) and the lower shear layer entering the gap, $c_g$ near the UC shifts downwards, and $\bar {F}_g$ reduces significantly. The connected top shear layer from the UC to the DC breaks apart by the lower shear layer advected upwards with gap flow. However, the dominant part of the UC top shear layer diffuses with the shear layer of the DC to form an extended top shear layer advecting downwards, and thereby separating the ACW vortex from the lower shear layer.

For $t_3$, as the UC approaches the lower extremum, a downwards $c_g$ is induced in the gap, resulting in a push on the top shear layer towards the UC. The low-pressure region on the top surface of the UC starts pulling back of the UC, resulting in a net dissipation of energy from the UC ($W_1<0$) around $t_3$. Simultaneously, the lower shear layer of the UC starts impacting the DC on the front surface, while being pushed downwards. This creates a low-pressure region on the front lower side of the DC, causing retardation of the cylinder ($W_2<0$) around $t_3$.

For $t_4$, as the cylinders move to their opposite extreme positions, a maximum of $\bar {F}_g$ pushes the lower shear layer of the UC downwards and attaches to the lower shear layer of the DC, while the top shear layer gets pushed into the gap. This forms a mirror image of the $t_0$ instant, and the cycle continues.

Overall, a smaller $\dot {y}_1$ at larger $U_R$ results in relatively farther proximity of the low-pressure core of the shear layer. Consequently, the induced gap flow due to the DC is responsible for pushing the shear layer closer to the UC and energizes the FIV at higher $U_R$. Similarly, the DC receives a lower mean flow due to the shielding effect of the UC. The shear layers of the DC cannot induce large $A_2$ by themselves. Therefore, the shear layers of the UC merge with shear layers of the DC to form strong vortices and attain large FIV amplitudes.

6.2.3. The LN regime: Case 2

The LN regime for Case 2 at $G = 3$, $U_R = 6.5$ is characterized by $y_1$ ($A_1 \sim 0.17$) significantly smaller than $y_2$ ($A_2 \sim 0.87$). The modal ratio for Mode 1 oscillation at this $U_R$ is $\sigma _{n1} \sim 0.19$, which is the ratio of FIV amplitudes. This implies that the mode shapes strongly influence the relative amplitude, especially during lock-in with that modal frequency. The $C_{T1}^\prime \sim 0.54$ is slightly smaller than $C_{T2}^\prime \sim 0.63$. Figure 12(e) shows a $C(2S)$ vortex shedding pattern in the downstream wake, similar to the LB regime of Case 1. Interestingly, while a typical gap flow excitation requires an out-of-phase motion of the cylinders (Borazjani & Sotiropoulos Reference Borazjani and Sotiropoulos2009), the Case 2 configuration involves an in-phase motion of the cylinders due to the elastic coupling and is discussed as follows.

Figure 12. Transient FIV characteristics of Case 2 LN regime. The rest of the caption is same as in figure 10. See also supplementary movies of transient wake animations for the LN regime of Case 2 (movie7.mp4).

Figure 12(b) shows a non-zero time average of $W_1$ and $W_2$, implying a net energy gain of the UC and net energy loss of the DC to the flow, even though there is no structural damping present in the system. This is due to the net transfer of the flow energy from the UC to the DC, through the elastic coupling. Further, the Case 2 LN regime is characterized by $A_2>A_1$, similar to Case 1 LB regime, which exhibits qualitative similarities, as seen by comparing figures 12 and 11. The transient $W_2$ variation is qualitatively similar in the two cases, with shear layers alternately separating from the UC and attaching to the corresponding shear layers on the DC (figure 12f at $t_3$). The gap flow between the two cylinders pushes the UC shear layer into the gap, pinching off the connected shear layer between the UC and the DC (at $t_0$). The associated pressure variations on the two cylinders result in an overall FIV mechanism, similar to the Case 1 LB regime.

However, there are some notable differences introduced into the FIV system due to the elastic coupling. As the UC and the DC move in-phase, a backward gap flow sets up near the UC (figure 12f at $t_0$). This causes a reduced magnitude of $\bar {F}_g$, compared with the corresponding Case 1 LB regime (figure 11f at $t_0$), and a distant low-pressure core of the shear layer behind the UC (figure 12f at $t_0$). Consequently, no energy is gained in the ($t_0, t_1$) interval. Furthermore, due to the downstream shifting of the strong forward gap flow towards the DC at $t_1$, the upper shear layer from the UC attaches DC farther downstream, resulting in a lower energy gain from the flow in ($t_1, t_2$). The lower shear layer from the UC impacts directly on the DC, causing a larger energy loss to the flow in ($t_2, t_3$). As the UC approaches maximum $\dot {y}_1$ at $t_2$, the upper shear layer pushes closer to the cylinder, resulting in a net energy gain from flow ($W_1>0$). The gained energy is transferred to DC through the elastic coupling for sustaining the FIV of the coupled system.

Overall, the flow energizes the DC through the UC via the elastic coupling. Therefore, the system in Case 2 is able to exhibit LB characteristics at much lower $U_R$, as compared with the Case 1 LB regime. However, the modification in the relative motion of the cylinders results in an inefficient gap flow between the cylinders, reflected by $F_g<0$ in ($t_1, t_2$).

6.2.4. The LN regime: Case 3

The LN regime for $G = 4$, $U_R = 9$ of Case 3 is characterized as follows. Figure 13(a) shows that $y_1$ ($A_1 \sim 0.89$) is slightly larger than $y_2$ ($A_2 \sim 0.73$), which is very close to the corresponding modal ratio $\sigma _2 \sim -1.26$. The negative sign represents out-of-phase motion, reflected by $\phi _{y12} \sim 180^{\circ }$. The $C_{T1}^\prime \sim 0.67$ is slightly smaller than $C_{T2}^\prime \sim 0.75$, with both signals being asymmetric across the two half-periods. This is also reflected in the ASD signatures (figure 13c,d) with even and odd harmonics of the lock-in frequency. Further, the even harmonics are stronger in $C_{T2}$, as compared with $C_{T1}$. The downstream wake has a characteristic $P+S$ vortex shedding pattern (figure 13e), with a vortex pair in the lower row and a single vortex in the upper row.

Figure 13. Transient FIV characteristics of the Case 3 LN regime. The rest of the caption is same as in figure 10. See also supplementary movies of transient wake animations for the LN regime of Case 3 (movie8.mp4).

Figure 13(b) plots a net energy gain (loss) at the DC (UC) from the flow, with a dominant energy transfer from the DC to the UC through the elastic coupling. Here the UC generates a gap flow wave similar to Case 1 UB regime, however, the overall FIV mechanism is similar to the Case 1 LB regime, discussed as follows. In figure 13(f), the lower extreme position of the UC at $t_0$ is marked by the upper shear layer being pushed downwards creating a low-pressure region on the upper side and absorbing energy from the flow ($W_1>0$). The DC also experiences a minor pressure drop due to the lower shear layer of the cylinder. As the UC moves upwards, the shear layer advects downstream and the front stagnation point moves upwards. However, the pressure drop induced by the upper shear layer is weak due to low $\dot {y}_1$ at high $U_R$, and is unable to overcome the downward push exerted on the UC by the stagnation pressure on the front surface in ($t_1, t_3$). As a result, the UC loses significant energy to the flow ($W_1<0$). Simultaneously, the lower shear layer of the UC, which passes through the gap at $t_0$, is forced downwards by the gap flow and reattaches with the lower shear layer of the DC. Subsequently, the lower shear layer of the DC grows stronger and the net pressure drop on the rear lower side of the DC increases drastically, pulling the DC downwards. This is also confirmed by $W_2>0$ in ($t_0, t_2$).

As the UC crosses the mean position and reaches $t_3$, $\dot {y}_1$ reduces, and the stagnation point moves back to the frontal surface, gradually reducing the rate of energy dissipation to the flow. The upper shear layer of the UC, advected by gap flow, reaches the lower side of the DC and separates the connected lower shear layers of the UC and the DC. As a result, the extended lower shear layer of the DC forms a smaller vortex farther downstream and a larger vortex closer to the DC, with the UC shear layer merging with the larger vortex at $t_4$. This larger vortex formation close to the DC is responsible for reintensifying the low-pressure region behind the DC, and a net energy gain of the cylinder system from the flow ($W_2>0$) around $t_3$.

While the near wake shear layers at $t_4$ almost mirror the $t_0$ configuration, the ACW vortex shed by the DC at $t_4$ is slightly stronger than the CW vortex at $t_0$. This is more evident from the pressure contours at $t_0$ and $t_4$. As discussed in § 6.2, this larger ACW vortex is able to merge downstream with the smaller ACW vortex formed at $t_3$, while the CW counterpart does not merge, resulting in a $P+S$ vortex shedding pattern. It should be noted that UC is unable to gain flow energy at $t_0$ or $t_4$ instant, due to the lower intensity of $c_g$ close to the UC. At smaller $G$ (see figure S10), the UC and the DC gain higher energy from the flow.

Overall, the UC dissipates energy into the flow by energizing the gap flow, confirmed by a maximum $c_g$ close to the DC at $t_2$. This gap flow elongates the shear layers, which eventually attach to DC. The merged shear layers of the two cylinders create strong vortices in downstream of the DC, resulting in large FIV amplitudes. The DC eventually transmits the gained energy from the flow to the UC for sustaining large $A_1$ and the resulting gap flow. Therefore, the LN regime of Case 3 is sustained at a much larger $U_R$ than the Case 1 LB regime due to the sustainable large gap flow, induced by the UC vibrations.

6.3. Galloping regime

Figure 14 plots the Case 1 GD regime for $G = 1.1$ at $U_R = 16$. As show in figure 14(a), $y_2$ ($A_2 \sim 1.1$) is larger than $y_1$ ($A_1 \sim 0.9$), with $\phi _{y12} \sim 326^{\circ }$. The ASD signatures in figure 14(c,d) show dominant frequencies of $y_1$, $y_2$ and those of $C_{T1}$, $C_{T2}$ closer to $f_{n1}$. While $St_{00}$ is close to the even harmonic of FIV vibration frequency, only odd harmonics are observed in the spectra of $C_{T1}$ and $C_{T2}$. Figure 14(b) indicates that the cylinders gain energy from the flow ($W_i>0$) while they move towards the mean position, and dissipate energy ($W_i<0$) as they move away from the mean position. Chen et al. (Reference Chen, Alam, Li and Ji2023) reported similar energy exchange characteristics in the galloping response of D-section cylinder at larger $U_R$.

Figure 14. Transient FIV characteristics of Case 1 GD regime. The rest of the caption is same as in figure 10. See also supplementary movies of transient wake animations for the GD regime of Case 1 (movie9.mp4).

The transient wake in figure 14(f) shows that the UC and the DC align themselves such that the DC is permanently submerged between the upper and lower shear layers of the UC. However, the DC is slightly misaligned from the UC wake centreline. As the UC moves upwards in ($t_0, t_2$), the DC follows the UC by slightly blocking the lower shear layer. This creates a low-pressure region on the upper side of the UC and the DC, which pulls both cylinders upwards. Further, a weak gap flow between the cylinders results in flow reattachment on the lower side of the UC, and downward shifting of the front stagnation point of the DC. This further results in an upward push to the cylinders from the lower side. As the UC crosses the mean position around $t_1$, $\dot {y}_1$ starts reducing while $\dot {y}_2$ increases. This results in the shifting of the DC towards the upper shear layer of the UC, and the formation of a low-pressure region on the lower side of the UC and the DC during ($t_2, t_3$). As the cylinders move upwards, this low-pressure region strongly retards the cylinders around $t_3$. At $t_4$, the UC moves downwards, with the low-pressure region on the lower side due to the upward shifting of the DC. Since the cylinder motion is governed by minor obstruction of shear layers of the UC by the DC, and not entirely by vortex–cylinder interactions, the system continues to gallop at large $U_R$.

The cyclic obstruction of the UC shear layers by the DC results in an uneven strength of elongated shear layers in the downstream, those later interact to form a $2P$ vortex shedding pattern (figure 14e). The dominant ASD frequency for both the cylinders in figure 14(c,d) is slightly larger than $f_n$, indicating a negative added mass. In this context, Horowitz & Williamson (Reference Horowitz and Williamson2010) reported that the $2P$ vortex shedding mode can induce a negative added mass, consistent with our observations here.

Case 3 galloping is described in figure 15 for $G = 1.1$ and $U_R = 16$. Contrary to Case 1, Case 3 galloping is characterized by $y_1$ ($A_1 \sim 0.73$) larger than $y_2$ ($A_2 \sim 0.55$), with $\phi _{y12} \sim 227^{\circ }$ (figure 15a). This is a significant deviation from the expected natural mode parameters, i.e. $\sigma _2 \sim -2.864$ and $\phi _{y12} \sim 180^{\circ }$. Further, the mean position of the DC also shifts downwards by $\bar {y}_2 \sim -0.14$. The $C_{T1}^\prime \sim 0.9$ and $C_{T2}^\prime \sim 0.7$ also show larger cycle-to-cycle fluctuations, compared with Case 1 GD. The ASD signatures in figure 15(c,d) show the presence of strong even harmonics in $C_{T1}$ and $C_{T2}$, along with the dominant FIV frequency and its odd harmonics. Traces of even harmonics are also reflected in the $y_1$ and $y_2$ signals. The dominant frequency of the signals is slightly smaller than $f_{n2}$ and indicates strong added mass effects. Additionally, figures 7(diii) and 7(div) show the frequency component branch is parallel to $f_{St_{00}}$, which merges with the even harmonic branch at higher $U_R$. Therefore, the first even harmonic peak in figure 15(c,d) corresponds to the modified vortex shedding frequency, similar to those observed for D-section cylinder galloping vibrations (Sharma et al. Reference Sharma, Garg and Bhardwaj2022a). The overall vortex shedding pattern is $T+P$, indicating strong correlation of $\bar {y}_2$ with the flow characteristics. The transient flow characteristics are plotted in figure 15(f) and are discussed as follows.

Figure 15. Transient FIV characteristics of the Case 3 GD regime. The rest of the caption is same as in figure 10. See also supplementary movies of transient wake animations for the LN regime of Case 3 (movie10.mp4).

For $t_0$, when the UC is at its upper extreme position and the DC returns from its lowermost position, a large gap flow is induced between the cylinders. The top shear layer of the UC does not reattach to the DC (as observed in the Case 3 LN regime), resulting in a CW vortex. The lower shear layer of the UC is weak and is pushed farther downstream by the gap flow, forming a weak low-pressure region on the lower side of the UC.

For $t_1$, the UC and the DC start moving rapidly towards each other, and thereby reducing the gap flow. As a result, the shear layers passing through the gap become weaker, increasing the pressure on the lower side of the UC. Further, as the UC is moving downwards, the front stagnation point moves downwards and the upper shear layer separates earlier, resulting in energy dissipation from the UC to the flow ($W_1<0$). While a similar phenomenon occurs when the DC moves upwards, the energy dissipation effects are not significant due to shielding effects on the upper side due to the UC, and upwards shifting of the front stagnation point attributed to gap flow. Simultaneously, the lower shear layer of the DC rolls up to form a ACW vortex, pushing the CW vortex downstream.

For $t_2$, as the cylinders cross the centreline, both shear layers of the UC pass around the DC with negligible gap flow. However, due to the relative motion of the two cylinders, the lower shear layer of the UC attaches to the DC, while the upper shear layer flows past the DC. The ACW vortex of the DC grows rapidly from the attached lower shear layer of the UC, and advects farther downstream.

For $t_3$, as the UC moves farther downwards, the DC blocks the upper shear layer while it pulls the attached lower shear layer upwards. This creates a high (low) pressure region on the upper (lower) side of UC, leading to a strong displacing force from the flow ($W_1>0$). While this orientation causes energy dissipation at the DC, this effect is reduced by the formation of a low-pressure region on the rear upper surface of the DC. The upper shear layer of the UC reattaches with the DC shear layer to form a CW vortex.

For $t_4$, the UC reaches the lowermost position with the DC, initiating its downwards motion. While this marks the half-time period of oscillation, the cylinder wake is significantly different from the expected mirror image. The CW vortex of the DC at $t_4$ is much larger than the ACW vortex of the DC at $t_0$. Further, the gap flow diminishes due to the downwards-shifted mean position of the DC. This results in an upstream shifting of the ACW vortex at $t_4$, with both the UC and the DC shear layers feeding to it. As a result, this ACW vortex grows stronger, compared with its CW counterpart at $t_0$. Further, the upper shear layer of the UC is more concentrated at $t_4$, resulting in a net energy gain from the flow ($W_1>0$).

For $t_5$, as the cylinders move towards the mean position, the stronger ACW vortex rapidly moves upwards and separates the CW vortex. Similar to $t_1$, the UC dissipates energy to the flow, while the energy dissipated by the DC in this interval is regained from the ACW vortex downstream.

For $t_6$, while the lower shear layers still feed the downstream ACW vortex at $t_2$, the upper shear layers of the UC and the DC combine to form another CW vortex due to the early separation of the CW vortex. However, this vortex results in a low-pressure region formation on the rear upper side of DC, causing a significant energy dissipation to the flow ($W_2<0$). At the same time, the upper shear layer of the UC, attached to the DC, gets pulled downwards with the DC. As a result, a strong low-pressure region is formed on the upper side of the DC in ($t_6, t_7$), which regains energy from the flow ($W_1>0$). This regained energy is transferred to the DC through the elastic coupling to sustain its downward motion.

For $t_7$, as the CW vortex separates from the DC around $t_7$, the upper shear layers of the UC and the DC combine to form another CW vortex. This results in a CW vortex, that advances to $t_0$.

Overall, the galloping mechanism of the UC also depends upon the tendency of the DC blocking either side of the UC shear layers. However, the DC does not sustain FIV in the out-of-phase motion with the UC, which is compensated by transferring energy from the UC to the DC via the elastic coupling. Further, larger amplitudes lead to the advection of shear layers through the gap, resulting in an ineffective galloping excitation. Therefore, we report self-limiting amplitudes during galloping FIV for Case 3, instead of continuously increasing FIV amplitudes in Case 1.

6.4. Initial branch and mixed mode regime

Data of Case 1 at $G = 5$ and $U_R = 4.75$ is plotted for the IB regime in figures 16(ai–16ax). The $y_1$ ($A_1 \in [0.33,0.36]$) is larger, and has smaller fluctuations than $y_2$ ($A_2 \in [0.12,0.25]$). Similarly, $C_{T1}^\prime \in [1.32,1.66]$ is also larger and has smaller fluctuations than $C_{T2}^\prime \in (0,1.68]$. Further, ASD of $y_1$ and $C_{T1}$ indicate a primary peak slightly lower than $f_{n1}$ (figures 16aii–16aiii), indicating positive added mass effects. The second dominant peak corresponds to $St_{00}$. While there are multiple peaks in the ASD plot, frequency values alone are inadequate to characterize the corresponding peaks. Therefore, DMD is performed to capture the different wake modes (Rowley et al. Reference Rowley, Mezić, Bagheri, Schlatter and Henningson2009; Schmid Reference Schmid2010; Jovanović, Schmid & Nichols Reference Jovanović, Schmid and Nichols2014). Figure 16 shows the corresponding DMD frequencies of the wake, with fundamental frequencies and their harmonics marked therein. The grey bars show harmonic interaction of $f_1$, $f_2$ and $f_3$. The reconstructed wake in figure 16(avi) from the decomposed frequencies (excluding the grey bars in figure 16aiv) shows good agreement with the actual wake in figure 16(av). The individual modal wakes (figures 16avii–16ax), obtained from DMD, show vortex coshedding regime (Zdravkovich Reference Zdravkovich1988) at $f_1$ similar to stationary tandem cylinders, i.e. corresponding to $St_{00}$. Figure 16(aviii) shows vortex shedding from the cylinders at $f_2$ closer to $f_n$, with ordered $C(2S)$ pattern similar to lock-in region. In addition, figure 16(aix) shows another vortex-shedding mode at $f_3$ similar to the $2S$ pattern observed in the ID regime, in the absence of gap vortices. Therefore, DMD evidently shows that the two vortex shedding modes (with and without gap vortex) and one structural mode coexist in the IB regime with different frequencies, resulting in a fluctuating quasiperiodic FIV response.

Figure 16. The FIV characteristics of the Case 1 IB (ai–ax) and MM (bi–bx) regime. Here $y_1$, $y_2$, $C_{T1}$ and $C_{T2}$ are plotted in (i). The ASD (without normalization) is plotted for $y_1$ and $C_{T1}$ in (ii), and for $y_2$ and $C_{T2}$ in (iii). The DMD modal amplitudes are plotted in (iv). Flow wake and reconstructed wake from the considered DMD modes are plotted in (v) and (vi), respectively. First three wake modes, formed by $f_0 + \sum f_i$ are plotted in (vii) ($i=1$), $8$ ($i=2$) and $9$ ($i=3$). The panels (x) show the $f_0 + \sum (\,f_2-f_1)$ mode.

The MM regime is observed only for Case 3 and is characterized using tandem cylinders with $G = 5$ at $U_R = 4.75$. Figure 16(bi) shows slightly larger $y_1$ ($A_1 \in [0.03,0.21]$) with larger fluctuations than $y_2$ ($A_2 \in [0.08,0.19]$). Similarly, $C_{T1}^\prime \in [0.52,0.67]$ is larger than $C_{T2}^\prime \in (0,1.75]$, with fluctuations larger on the DC. The ASD signatures show a dominant peak at $f_1 \sim f_{n2}$ for the UC and $f_2 \sim St_{00}$ for both the UC and the DC. Further, a dominant peak is also observed at $f_3 \sim f_{n1}$ (seen clearly in figure 6), corresponding to the in-phase vibration mode. However, $f_3$ is shifted to a higher frequency, indicating a negative added mass. In this context, Bao et al. (Reference Bao, Huang, Zhou, Tu and Han2012) studied the 2-DOF vibration of tandem cylinders at $Re=150$ with varying frequency ratios between the in-line and transverse directions. They observed strong negative added mass effects for the in-line vibrations, that drive the natural frequency closer to the flow excitation frequency. Similarly, a negative added mass is induced for in-phase vibration mode, and thereby $f_3$ moves close to $f_2 \sim St_{00}$. Similar to the IB regime, the DMD for the MM regime figure 16(biv) shows dominant wake modes corresponding to dominant ASD frequencies. Figures 16(bvii), 16(bviii) and 16(bix) correspond to the stationary tandem cylinder coshedding (Mode 2) at ${\sim }St_{00}$, out-of-phase structural vibration (wake Mode 1) at $\sim f_{n2}$ and stationary tandem cylinder reattached flow (wake Mode 3), respectively. However, more distinct vortices are formed for wake Mode 2 (figure 16bviii) and wake Mode 3 (figure 16bix), as compared with the corresponding IB regime. This implies a stronger FSI coupling between the structural natural frequency and the wake modes. As the wake Mode 3 is significantly weaker in the MM regime, we observe a dominant quasiperiodic coshedding regime.