3.1. Mathematical model

For the present FmSI problem, the fluid dynamics is modelled by the Navier–Stokes equations and the vibrational dynamics (of the elastically mounted cylinder) is modelled by a linear oscillator equation. The fluid–structure interface dynamics is governed by an interface boundary condition. Non-dimensional form of the governing Navier–Stokes equations are given as

(3.1) \begin{equation} \nabla .\boldsymbol{V}^{*}=0, \end{equation}

(3.2) \begin{equation} \frac {\partial \boldsymbol{V}^{*}}{\partial t^*}+(\boldsymbol{V}^{*}.\nabla ) \boldsymbol{V}^{*}=-\nabla p^*+ \frac {1}{Re} {\nabla }^2 \boldsymbol{V}^{*}, \end{equation}

where $\boldsymbol{V}^{*}=({\boldsymbol{u}}/u_{\infty })$ is the non-dimensional velocity vector, $p^*=(p/ \rho _f {u_\infty }^2)$ is the non-dimensional pressure and $Re(=\rho _f u_\infty D/{\mu })$ is the Reynolds number. Here, the dimensional variables $\boldsymbol{u}$ , $p$ , $\rho _f$ and $\mu$ represent velocity, pressure, fluid density and viscosity, respectively.

For the vibrational dynamics, the governing linear spring–mass–damper oscillator equation is given as

(3.3) \begin{equation} \ddot {y^*}+ \frac {4\pi \zeta }{U^*} \dot {y^*}+\left (\frac {2\pi }{U^*}\right )^2 y^* = \frac {2C_{l}}{\pi m^*}, \end{equation}

where $\ddot {y^*}$ , $\dot {y^*}$ and $y^*$ represent non-dimensional acceleration, velocity and displacement, respectively. Further, $C_l=F_y/(({1}/{2}){\rho }_f {u_\infty }^2 D)$ is the lift coefficient, $U^* = U_{\infty }/(f_n D)$ is the reduced velocity, $\zeta$ is the damping ratio and $m^* = m_c/m_d$ is the mass ratio. Here, the dimensional parameters $f_n$ , $m_c$ and $m_d$ are natural frequency, mass of cylinder and mass of fluid displaced by the cylinder, respectively. Also, $F_y$ is the transverse force acting on the oscillating cylinder.

At the fluid–structure interface, the solid and fluid velocity should be continuous; given (Mittal & Sharma Reference Mittal and Sharma2022) as

(3.4) \begin{equation} \boldsymbol{V}^*_{s,int} = \frac {dy^*}{dt} = \boldsymbol{V}^*_{f,int}, \end{equation}

where $\boldsymbol{V}^*_{s,int}$ and $\boldsymbol{V}^*_{f,int}$ are non-dimensional solid and fluid velocity at the interface, respectively.

3.2. Numerical methodology

The present work uses a levelset function-based immersed interface method (LS-IIM) that was proposed in our research group (Thekkethil & Sharma Reference Thekkethil and Sharma2019). A physical law-based finite volume method (Sharma Reference Sharma2022) is used for the flow solver. Further, the LS-IIM solver uses a timewise fixed, non-uniform and non-body-fitted Cartesian grid for the flow solver. The present method uses a levelset function-based implementation of the interfacial boundary conditions. The flow solver uses the Quadratic Upwind Interpolation for Convective Kinematics scheme for the advection term and the central difference scheme for the diffusion term. Solution algorithm for the present flow solver can be found in the work of Thekkethil & Sharma (Reference Thekkethil and Sharma2019, Reference Thekkethil and Sharma2020).

Figure 2.

Non-uniform Cartesian grid distribution throughout the computational domain with uniform fine grid size of 0.025. The inset figure present enlarged view of the mesh around the cylinders.

3.3. Code validation and grid independence study

For the present study, figure 2 shows a Cartesian grid that consists of a uniform fine grid $\delta$ near the structure, a uniform coarse-grid $\varDelta$ near the outer boundaries and a hyperbolic stretching function-based non-uniform grid in the intermediate region. Table 1 presents a grid independence study at $G^* = 0.5$ and $U^* = 10$ – corresponding to almost maximum amplitude $A^*$ found in the present work. After comparing the results in the table, the mesh $M_2$ (with $\delta = 0.025$ and $\Delta t = 0.005$ ) is considered sufficiently small for accurate results, and used in the present simulations.

Table 1.

The FIV of an elastically mounted cylinder with a stationary cylinder downstream at $Re = 100$ , $m^*\,{=}\,2$ , $\zeta = 0.005$ , $G^* = 0.5$ and $U^* = 10$ , with $\delta$ as the uniform fine grid near the cylinder and $\Delta t^*$ as the time step. Further, the non-dimensional results are presented here with $A^*$ as vibrational amplitude, $f^*$ as oscillation frequency, $C'_{l}$ as root-mean-square lift coefficient and $\bar {C_d}$ as mean drag coefficient.

The present LS-IIM based in-house code has been validated for several benchmark problems on FSI by Thekkethil & Sharma (Reference Thekkethil and Sharma2019), and used for hydrodynamic investigations of fish-like locomotion of hydrofoils by Thekkethil et al., (Reference Thekkethil, Sharma and Agrawal2018, Reference Thekkethil, Sharma and Agrawal2020). Further, the in-house code was used for FmSI of a fixed or elastically mounted cylinder with a downstream flexible splitter plate in the work of Mittal & Sharma (Reference Mittal and Sharma2021, Reference Mittal and Sharma2022). In the present work, further validation study is presented for two problems: first, flow across two stationary cylinders in tandem arrangement; second, VIV of an elastically mounted isolated cylinder. Good agreement between the present and published results is shown in table 2 for the first and figure 3 for the second problem. Further, for the second problem, the present vibrational amplitude $A^*$ is closer to the result of Borazjani and Sotiropoulos (Reference Borazjani and Sotiropoulos2009), with a deviation of less than 4.3 %.

Table 2.

For flow over two stationary cylinders in tandem arrangement at various gap $G^*$ and $Re = 200$ , present and published results for mean drag coefficient $\bar {C_d}$ , amplitude of lift coefficient $C_{l,max}$ and Strouhal number $St$ of the two cylinders. Here, subscripts 1 and 2 represent the upstream and downstream cylinders, respectively.

Figure 3.

For VIV of an elastically mounted isolated cylinder at $Re = 150$ , $m^* = 2$ and $\zeta = 0$ , present and published results for variation of vibrational amplitude $A^*$ with increasing reduced velocity $U^*$ .

3.4. Parametric details

For the effect of the downstream cylinder-induced proximity interference on FIV of an elastically mounted cylinder, the present parametric study considers various values of the cylinder–cylinder gap $G^*(=0.1{-}2.5)$ and reduced velocities $U^* (= 3{-}20)$ , and a constant value of the other non-dimensional governing parameters; given as gap ratios $G^*$ : 0.1, 0.3, 0.5, 1.0, 1.5, 2.0, 2.5, reduced Velocity $U^*$ : 3–20 at a constant Reynolds number $Re = 100$ , mass ratio $m^* = 2$ and damping ratio $\zeta = 0.005$ .

The effect of the above non-dimensional input parameters on vibrational characteristics is presented in terms of non-dimensional amplitude $A^*$ , time-averaged or mean displacement $\bar {y^*}$ , frequency $f^*$ , phase difference $\phi$ (between the displacement $y^*$ and lift coefficient $C_{l}$ of the oscillating cylinder) and instantaneous non-dimensional power $\beta$ given as

(3.5) \begin{align} A^* &= (y^*_{max}\,{-}\,y^*_{min})/2;\ \;\bar {y^*}\,{=}\,\frac{1}{nT^*}\!\int _{t^*}^{t^*+nT^*}\! y^* {\rm d}t^*;\ \; f^*\,{=}\,\frac {f_y}{f_n};\ \;f^*_{st}\,{=}\,\frac{f_{st}}{f_n};\ \ \beta = 0.5 C_{l} \dot {y^*}, \end{align}

where $y^*_{max}$ is the maximum and $y^*_{min}$ is the minimum non-dimensional displacement. Further, $f_y$ is oscillation frequency, $f_n$ is the natural frequency of the cylinder, $f^*_{st}$ is the non-dimensional frequency of vortex shedding of an upstream stationary cylinder (for two cylinders in tandem arrangement) and $\dot {y^*}$ is the velocity of the cylinder. Also, $T^*$ is the non-dimensional time period of an oscillation cycle and $n$ is total number of cycles – considered for the time averaging.

Further, coefficients of pressure $C_p ({=}P/({1}/{2})\rho _f U^2_{\infty } D)$ and skin friction $C_f ({=}\tau/({1}/{2})\rho _f U^2_{\infty } D)$ ) around the surface of the cylinder are computed – to present the angular position $\theta$ of separation points and the front stagnation point on the cylinder surface. Furthermore, the total lift coefficient $C_l$ is presented by considering two components (Williamson & Govardhan Reference Williamson and Govardhan2004; Sharma et al. Reference Sharma, Garg and Bhardwaj2022): a potential lift coefficient $C_{l,p} ({=}{\kern-2pt}-{\kern-2pt}2C_a S \ddot {y^*}$ ) that is attributed to potential added mass of the cylinder, and a vortex lift coefficient $C_{l,v} (=C_l-C_{l,p})$ originating from the vorticity dynamics. Here, for the circular cylinder with diameter $D=1$ , the potential added mass coefficient $C_a=1$ and non-dimensional cross-sectional area $S=\pi /4$ . This dissection of $C_l$ acknowledges the construction of a flow solution by combining a potential flow field with a velocity field linked to vorticity.

For two structures in tandem arrangement, previous FIV studies (Zdravkovich Reference Zdravkovich1988; Borazjani & Sotiropoulos Reference Borazjani and Sotiropoulos2009; Mittal & Sharma Reference Mittal and Sharma2022; Sharma & Bhardwaj Reference Sharma and Bhardwaj2023) indicated a strong dependence of FIV response on qualitative characteristics of the gap flow – flow within the gap between the two structures. In the presents study, we recognised the importance of quantifying the gap flow in terms of volumetric flow rate $Q^*_g$ (within the cylinder–cylinder gap), and presented a comprehensive analysis of gap-flow characteristics and their effect on the various types of FIV responses.

Figure 4.

Schematic to illustrate the present methodology, for computation of non-dimensional gap-flow rate $Q^*_g$ – along the line joining the two cylinders at two different positions of the vibrating cylinder.

For computation of $Q^*_g$ , called hereafter the gap-flow rate, figure 4 presents a schematic representation of $Q^*_g$ that is dependent on the relative position of the two cylinders. For an instantaneous position $y^*$ of the upstream cylinder when the leading edge of the downstream cylinder is exposed to the free-stream flow, figure 4(a) shows that $Q^*_g$ is sum of the streamwise $Q^*_1$ and transverse $Q^*_2$ flow rates along the vertical ( $\Delta y^*$ ) and horizontal ( $\Delta x^*$ ) lines, respectively. Whereas, for the other $y^*$ , figure 4(b) shows that $Q^*_g$ is equal to the vertical flow rate $Q^*_2$ through the horizontal line $\Delta x^*$ . Also, note from the figure that the instantaneous $Q^*_g$ is defined such that it considers the flow across a surface that corresponds to a line joining the cell centres of the two cylinders; shown as a red line in the figure. Further, $Q^*_g$ oriented in the upward and downward directions in the gap are considered as positive and negative, respectively.

4.1. The FIV characteristics of an isolated cylinder

For the no proximity interference case of an isolated cylinder, with increasing $U^*$ , figure 5 shows the well reported increasing–decreasing trend of $A^*$ and $C_{l,rms}$ , along with an in-phase to anti-phase jump in the phase difference $\phi /\phi _{v}$ (between the temporal variation of $C_l/C_{lv}$ and cylinder displacement $y^*$ ). Further, the figure shows a demarcation of the FIV response into three branches – initial branch (IB, $U^* = 3.5{-}4.0$ ), upper branch (UB, $U^* = 4.5{-}5.5$ ) and lower branch (LB, $U^* = 6{-}8$ ) – in between the initial and final de-synchronisation regimes at small and large $U^*$ , respectively. These branchwise FIV response along with criteria for the demarcations are well reported in the literature for the isolated cylinder (Govardhan & Williamson Reference Govardhan and Williamson2000; Sharma et al. Reference Sharma, Garg and Bhardwaj2022) and are presented here in table 3. Note that the work of Govardhan & Williamson (Reference Govardhan and Williamson2000) illustrated the difference between the two-branch (IB, LB) response at high mass damping ( $m^* \zeta$ ) and three-branch response (IB,UB,LB) at low $m^* \zeta$ , where upper branch is associated with $0^\circ$ to $180^\circ$ transition in the vortex phase ( $\phi _{v}$ ) along with an amplitude jump. This amplitude jump in UB is reported for low mass-damping systems at a relatively high $Re$ ( $3500{-}10\,000$ ). Whereas, for the low $Re$ ( ${\lt } 200$ ) in the present as well as published studies, a continuous amplitude response has been observed for the UB (Prasanth & Mittal Reference Prasanth and Mittal2008; Sharma et al. Reference Sharma, Garg and Bhardwaj2022).

Figure 5.

For VIV of an isolated cylinder at $Re = 100$ , $m^*=2$ and $\zeta = 0.005$ , variation of non-dimensional amplitude $A^*$ , frequency $f^*$ , phase $\phi _{v}$ between displacement and vortex lift coefficient $C_{l,v}$ , phase $\phi$ between displacement and total lift coefficient $C_{l}$ , r.m.s. value of the lift coefficients ( $C_{lv,rms}$ , $C_{l,rms}$ ) with increasing reduced velocity $U^*$ . Here, $f^*_{st}$ is the vortex shedding frequency of the stationary case. Further, ID: initial de-synchronisation, IB: initial branch, UB: upper branch, LB: lower branch, and FD: final de-synchronisation.

Table 3.

For isolated and proximity interference-based vibrations of a circular cylinder at smaller damping ratio $\zeta \lesssim 0.05$ , characteristics of the trends and values of the various parameter-based branchwise demarcations of the FIV response. Note that the lift and drag coefficients are for the upstream vibrating cylinder.

For the initial and final de-synchronisations, note from the table that the amplitude $A^*\,{\approx}\,0$ while the frequency ratio $f^*$ of the vibrating cylinder is close to the vortex shedding frequency $f^*_{st}$ of the stationary cylinder. Further, the table shows a rapid increase in $A^*$ , $C_{l,rms}$ and $C_{lv,rms}$ for the IB while the UB corresponds to a decreasing trend in $A^*$ , $C_{l,rms}$ and $C_{lv,rms}$ . The lock-in region consists of UB and LB, where the oscillation frequency shows a constant slope smaller than $f^*_{st}$ and remains close to the natural frequency with $f^* \approx 1$ ; marked as the lock-in region in figure 5. Note from the figure that $f^*=f^*_{st}$ in the Initial de-synchronization branch while $f^* \approx 1$ in UB and LB – corresponding to the lock-in region. The transition from $f^*=f^*_{st}$ to $f^* \approx 1$ occurs in the intermediate IB, leading to a beats-like temporal signal in displacement and lift of the cylinder (Prasanth & Mittal Reference Prasanth and Mittal2008; Soti et al. Reference Soti, Zhao, Thompson, Sheridan and Bhardwaj2018; Sharma et al. Reference Sharma, Garg and Bhardwaj2022). Further, for the IB, the dominant frequency is $f^*=f^*_{st}$ and $f^* \lt 1$ , i.e. non-lock-in. Thus, the IB is not considered here as a part of lock-in; consistent with the previous studies (Soti et al. Reference Soti, Zhao, Thompson, Sheridan and Bhardwaj2018; Sharma et al. Reference Sharma, Garg and Bhardwaj2022). The lock-in begins at the UB with single dominant frequency around $f^*\approx 1$ and onset of the $\phi_{v}$ transition from ${\sim}0^\circ$ to ${\sim}180^\circ$ . Further, within UB, the onset of the $\phi _{v}$ jump at $U^* = 4.5$ is accompanied by a decrease in $A^*$ , $C_{l,rms}$ and $C_{lv,rms}$ . In the LB, $\phi _{v} \sim 180^\circ$ and $\phi$ undergoes the $0^\circ -180^\circ$ phase transition – with $\phi = 180^\circ$ demarcating the end of the lock-in region. It must be noted from the figure that the critical $U^* = 5$ and $U^* = 6.5$ , for phase-jump in $\phi _{v}$ and $\phi$ , corresponds to the minimum $C_{lv,rms}$ and $C_{l,rms}$ , respectively. It is interesting to notice from figure 5 that as $A^*$ reaches maxima in the IB, the vortex force transits to out of phase $\phi _{v} \sim 180^\circ$ (restoring in nature) that leads to the onset of decreasing trends in both $C_{l,rms}$ and $A^*$ . This suggest that the $\phi _{v}$ jump is the mechanism to self-limit the maximum amplitude of vibration in VIV, as reported by Prasanth & Mittal (Reference Prasanth and Mittal2008). Further, this is attributed to a gradual change in the timing of vortex formation which governs the top and bottom suction regions of the cylinder in a oscillation cycle.

4.2. Amplitude response of an elastically mounted cylinder under the effect of proximity interference

The VIV of an isolated cylinder serves as a base case with no proximity interference, and results in an increasing–decreasing $A^*$ -response; shown in figure 5(a) and also by the dashed line in figure 6(a). This trend of the $A^*$ versus $U^*$ variation is called VIV amplitude response.

Introducing a stationary cylinder downstream of an elastically mounted cylinder leads to proximity interference that increases with decreasing gap $G^*$ . Figure 6(a) shows that this proximity interference leads to modifications in the amplitude $A^*$ -response as compared with the VIV $A^*$ -response for an isolated cylinder. The modifications are seen at intermediate and larger values of $U^*$ (smaller stiffness of the mounted spring).

For larger gaps $G^* = 2.5{-}1.5$ , figure 6(a) shows a more gradual decay along with an increase in the proximity interference-induced enhancement in the asymptotic value of $A^*$ at larger $U^*$ . Further, for the larger gaps $G^* \geqslant 1.5$ , it is interesting to note from the figure that the deviation in $A^*$ begins at $U^* = 8$ – corresponding to the end of LB and lock-in region of VIV response (figure 5). Bokaian & Geoola (Reference Bokaian and Geoola1984a ) referred to this response at the larger gap as vortex resonance due to the phenomenon of vortex lock-in. However, as compared with vortex resonance or VIV of an isolated cylinder, figure 6(a) shows a significant modification in amplitude response caused by the proximity interference. Therefore, the amplitude response at larger $G^* \geqslant 1.5$ is called here proximity-induced modified VIV (PImVIV). Further, various studies with a downstream cylinder have referred to this amplitude response differently (Kim et al. Reference Kim, Alam, Sakamoto and Zhou2009; Alam et al. Reference Alam and Meyer2013; Nishi et al. Reference Nishi, Ueno, Nishio, Quadrante and Kokubun2014).

With further decrease in the gap $G^* \leqslant 1.0$ , figure 6(a) shows a significant deviation from the VIV response – indicating a substantial enhancement in the proximity interference effect. As compared with the PImVIV response at larger $G^* \geqslant 1.5$ , the figure for $G^*= 0.5$ shows a rapid-increase plateau type of $A^*$ – called in the literature the proximity-induced galloping (PIG) response (Bokaian & Geoola, Reference Bokaian and Geoola1984a ; Mittal & Sharma Reference Mittal and Sharma2022). Whereas, at an intermediate gap $G^* = 1.0$ , the figure shows a blend of PImVIV and PIG responses – called here the transitional PImVIV–PIG response. For $G^* = 1.0$ , the figure shows a rapid increase in $A^*$ for $U^* \leqslant 5$ , then decreases slightly at $U^* = 6$ , followed by slight increase until $U^* = 8$ (similar to PIG response) and finally a much more gradual decrease of $A^*$ at larger $U^*$ (similar to VIV response). Further, for the transitional response at $G^* = 1.0$ , notice that the asymptotic value of $A^*$ is much larger as compared with those in the PImVIV cases at $G^* \geqslant 1.5$ . The FIV characteristics and flow physics occurring during this transitional response are missing in the literature.

Figure 6.

Effect of non-dimensional gap $G^*$ on variation of non-dimensional (a) vibrational amplitude $A^*$ and (b) mean value of displacement $\bar {y^*}$ of the upstream cylinder, with increasing reduced velocity $U^*$ . The schematic at top shows characteristics curve for VIV, PIG and combined PIG–PISV response at larger, intermediate and smaller $G^*$ . Further, $\bar {y^*} = 0$ for $G^* \geqslant 1.0$ and is not shown in (b).

For a much smaller gap $G^* = 0.3$ and 0.1, figure 6(a) shows the PIG response upto a critical $U^*_c$ and further increase in $U^*$ leads to a sudden drop in the amplitude. Also, figure 6(b) shows that the sudden drop in $A^*$ is accompanied by a non-zero mean displacement $\bar {y^*}$ of the cylinder as compared with $\bar {y^*} \approx 0$ for the two-sided symmetrical vibration (about the downstream cylinder) of the upstream cylinder. In these cases, the cylinder gets stuck on one side of the mean position and oscillates with a very small amplitude $A^*$ ; and almost corresponds to flow across two stationary cylinders in staggered arrangement. Thus, these non-zero $\bar {y^*}$ cases at larger $U^*$ for smaller $G^* \leqslant 0.3$ are called here proximity-induced staggered vibration (PISV) while the overall $A^*$ -response is called combined PIG–PISV. For a symmetric tandem arrangement of two cylinders in close proximity, the staggered vibrations had been reported for the first time in the literature. Figure 6(b) shows an increasing trend of the staggered mean position $\bar {y^*}$ with increasing $U^*$ , along with a larger value of $\bar {y^*}$ for the much smaller $G^* = 0.1$ as compared with that for $G^* = 0.3$ . For the much larger $\bar {y^*}$ cases at the smallest $G^* = 0.1$ , it is interesting to note from figure 6(a) that $A^*$ of the staggered vibration is almost equal to that for an isolated cylinder case at $U^* \geqslant 14$ . Thus, at sufficiently larger value of the offset or $\bar {y^*}$ of the one-sided staggered vibration, the proximity effect is negligible and the upstream cylinder begins to oscillate similar to FIV of an isolated cylinder.

Figure 7.

Transient behaviour of displacement $y^*$ of the cylinder during transition from symmetric-to-staggered vibration with increasing reduced velocity $U^*$ at (a) $G^* = 0.3$ and (b) $G^* = 0.1$ .

Figure 7 presents a transient behaviour of the transition from the two-sided symmetric vibration to one-sided staggered vibration at the smaller $G^* = 0.3$ and $0.1$ . For $G^*\,{=}\,0.3$ , figure 7(a) shows two-sided vibration at $U^* = 17.5$ and one-sided vibration at larger $U^* = 19$ . While, at intermediate $U^* = 18.5$ , a transient switching in the mean position $\bar {y^*}$ – from $\bar {y^*}\gt 0$ to $\bar {y^*}\lt 0$ – is seen for the one-sided staggered vibration. Such transient switching is not seen in figure 7(b) for the smaller gap $G^* = 0.1$ , with the two-sided vibration at $U^* = 9.7$ and the one-sided vibration at $U^* = 9.8$ . Due to a stronger influence of a proximity pressure (at the front stagnation point of the downstream cylinder) at smaller $G^*$ , the symmetric-to-staggered transition with $U^*$ is significantly sharper – within $U^* = 9.7{-}9.8$ for $G^* = 0.1$ as compared with $U^* = 17.5{-}19$ for $G^* = 0.3$ (figure 7).

The proximity interference effect on the vibration of an elastically mounted upstream cylinder has been studied for various types of downstream stationary structures: circular cylinder (Bokaian & Geoola, Reference Bokaian and Geoola1984a ; Kim et al. Reference Kim, Alam, Sakamoto and Zhou2009), vertical square plate (Su et al. Reference Su, He, Zhang and Feng2022) and splitter plate (Zhu et al. Reference Zhu, Li and Wang2020; Mittal & Sharma Reference Mittal and Sharma2022; Mittal et al. Reference Mittal, Abhishek and Sharma2023). These studies also reported the presently found transition from the PImVIV to PIG amplitude response – at closer proximity of the downstream structure. However, the present cylinder–cylinder system has several differences in vibrational response as compared with the cylinder–plate system (presented recently from our research group; see Mittal & Sharma Reference Mittal and Sharma2022 and Mittal et al. Reference Mittal, Abhishek and Sharma2023). For the present PImVIV response at larger gaps $G^*\geqslant 1.5$ , the present case in figure 6(a) shows larger peak amplitude $A^*_{max}$ (at $U^* = 5$ ) as compared with the isolated cylinder case while the cylinder–plate system leads to an almost same value of $A^*_{max}$ (Mittal & Sharma Reference Mittal and Sharma2022). Thus, at larger $G^*$ , the $A_{max}$ of the cylinder is enhanced by the proximity of a cylinder here but not affected by proximity of the splitter plate. At larger  $U^*$ , for the cylinder–plate system, Mittal & Sharma (Reference Mittal and Sharma2022) reported monotonic increase in $A^*$ with decreasing $G^*$ while such variation is observed here only up to $G^* = 0.5$ – due to onset of staggered vibration for $G^* \leqslant 0.3$ . The above discussed difference in the vibrational characteristics for the cylinder–cylinder and cylinder–plate systems suggests a substantial difference in the proximity interference mechanism – for a downstream bluff body (cylinder) here as compared with that for the more streamlined splitter plate.

For the cylinder–cylinder system, with decreasing $G^*$ , the experimental studies conducted by Bokaian & Geoola (Reference Bokaian and Geoola1984a ) ( $Re = 1150{-}5950$ ) and Kim et al. (Reference Kim, Alam, Sakamoto and Zhou2009) ( $Re = 4365{-}74\,200$ ) reported a similar amplitude transition of PImVIV to PIG. Nishi et al. (Reference Nishi, Ueno, Nishio, Quadrante and Kokubun2014) ( $Re = 2480{-}15\,000$ ) also reported a significant enhancement in the hydro-kinetic energy extraction at constant gap $G^* = 0.3$ , as compared with an isolated cylinder. Further, for PIG response, similar to the decrease of plateaued amplitude $A^*$ (at larger  $U^*$ ) with decreasing $G^*\leqslant 0.5$ (figure 6 a), Bokaian & Geoola (Reference Bokaian and Geoola1984a ) also reported a reduction in the plateaued PIG-response amplitude with decreasing $G^*$ from 0.25 to 0.09. However, they did not report staggered vibrations probably due to the smaller range of $U^*(\leqslant 6)$ considered in their experimental study. Su et al. (Reference Su, He, Zhang and Feng2022) also shows a similar decrease in amplitude $A^*$ of the cylinder with decreasing $G^*\leqslant 0.3$ , under the effect of a downstream vertical square plate (of dimensions $1.25D \times 1.25D \times 0.13 D$ ). However, due to the smaller spanwise length of the square plate as compared with the very long cylinder here, no staggered vibrations were reported for the square plate, probably due to spanwise leakage of the flow in between the cylinder and plate. Further, at $G^* = 0.3$ , Kim et al. (Reference Kim, Alam, Sakamoto and Zhou2009) also did not report staggered vibration due to the large mass damping of the vibrating system. Similar to Bokaian & Geoola (Reference Bokaian and Geoola1984a ), with decreasing $G^*$ , the PIG vibration response here commences at larger $U^* = 4$ for $G^* \geqslant 1.0$ , $U^* = 5$ for 0.5 and 0.3 and $U^* = 6$ for 0.1 in figure 6(a) for the downstream cylinder case here while it remains almost same for a downstream splitter plate case (Mittal & Sharma Reference Mittal and Sharma2022). The above studies at low and high $Re$ show consistent trends in vibrational responses with decreasing $G^*$ – along with shifts in the $G^*$ for the onset of transition between the vibrational responses. This highlights the relevance of the present low Re-based laminar flow findings and proposed proximity interference-induced flow mechanisms to turbulent flows at higher Reynolds numbers.

Figure 8.

For each of the four types of vibrational responses, with increasing reduced velocity $U^*$ , variation of non-dimensional ( $a1{-}d1$ ) amplitude $A^*$ and frequency $f^*$ , ( $a2{-}d2$ ) total phase $\phi$ and vortex phase $\phi _v$ (between displacement and total lift and vortex lift force) and ( $a3{-}d3$ ) r.m.s. value of total lift coefficient $C_{l,rms}$ and vortex lift coefficient $C_{lv,rms}$ for the vibrating cylinder. Further, $f^*_{st}$ is the vortex shedding frequency of the corresponding stationary configuration. Note that the results for the isolated cylinder are also presented with the dotted lines. Here, PIV: proximity-induced vibrations, PIG: proximity-induced galloping, and PISV: proximity-induced staggered vibrations.

4.3. The FIV characteristics of an elastically mounted cylinder with proximity interference from downstream cylinder

For each of the four amplitude responses, presented above, figure 8 shows vibrational characteristics separately for a representative case of the PImVIV, transitional PImVIV–PIG, PIG and combined PIG–PISV at $G^* = 2.0,1.0,0.5$ and 0.1, respectively. Similar to the branchwise demarcation of the FIV response for an isolated cylinder in figure 5, recently Sharma & Bhardwaj (Reference Sharma and Bhardwaj2023) presented such demarcations for two elastically mounted cylinders in tandem arrangements. They reported ID, IB, UB, LB, wake-induced vibration (at larger $G^* \geqslant 3$ ), FD (at intermediate $G^* = 1.0$ ) and a galloping-dominated regime (at smaller $G^* = 0.1$ ) with increasing $U^*$ . They also reported changes in the range of various FIV branches with decreasing $G^*$ . Whereas, for the present proximity-induced vibrations of the upstream cylinder, table 3 shows the characteristics of various branchwise demarcations of the vibrational characteristics.

Out of five branches (ID, IB, UB, LB and FD) in the vibrational characteristics for isolated cylinder in figure 5, the proximity interference-based vibrational characteristics in figure 8 show four branches (ID, IB, UB, LB) at larger $G^* \geqslant 1.0$ , two branches (ID, IB) at intermediate $G^* = 0.5$ and only one branch (ID) at smaller $G^* = 0.1$ . Moreover, at larger $U^*$ , the figure shows additional branches: proximity-induced vibration (PIV) at $G^* \geqslant 1.0$ , PIG at $G^* = 0.5$ and PIG along with PISV at $G^* = 0.1$ . For PImVIV and the transitional response, note from the figure that the UB and LB are characterised by in-phase to anti-phase transition in $\phi _{v}$ and $\phi$ , respectively, similar to that for the isolated cylinder in figure 5. These phase jumps do not occur for PIG response; thus, UB and LB are not seen in figure 8(c,d). For an isolated cylinder, a unique relationship is found between $f^*$ and $f^*_{st}$ , with $f^* \gt f^*_{st}$ in the IB/UB while $f^* \lt f^*_{st}$ in the LB for circular cylinder (figure 5) and vice versa (with no UB) for a square cylinder (Lin & Alam Reference Lin and Alam2024). Whereas, for the present case, where $f^*_{st}$ corresponds to that of two stationary cylinders, such a unique relationship is not found (figure 8) and is thus, not presented in table 5 for the branchwise demarcation.

For PImVIV and transitional response at larger $G^* \geqslant 1.0$ , figure 8(a,b) shows that the PIV branch of vibrational characteristics corresponds to larger asymptotic values of $A^*$ , $C_{l,rms}$ and $C_{lv,rms}$ as compared with those for the FD branch of the isolated cylinder (figure 8 and table 3). Further, the figure shows that the increasing $f^*$ (parallel to the $f^*_{st}$ ) is smaller for proximity interference cases as compared with that for the isolated cylinder, with $f^* \gt f^*_{st}$ in the PImVIV response and $f^* \lt f^*_{st}$ in the transitional response (figure 8 a,b). Additionally, figure 8(a,b) shows a smaller $C_{lv,rms}$ for the transitional ( $G^*\,{=}\,1.0$ ) as compared with the PImVIV ( $G^* = 2.0$ ) response – indicating decreased strength of the anti-phase force-generating vortex with decreasing $G^*$ .

The PIG branchwise response in figure 8(c,d) is characterised by an asymptotic increase (decrease) in $A^*$ ( $C_{l,rms}$ and $C_{lv,rms}$ ), along with an almost constant value of $f^*$ . The characteristics of the PISV branch are similar to the FD branch of an isolated cylinder (table 3) except for the mean displacement $\bar {y^*}\gt 0$ . Further, as VIV effects of the isolated cylinder dominate at smaller $U^*$ ( $\leqslant 6$ ), $\phi \approx 0^\circ$ for all gaps. As $U^*$ increases, proximity interference becomes more dominant over the VIV effects. Thus, with decreasing $G^*$ at larger $U^*$ ( $\geqslant 12$ ), $\phi \approx 180^\circ$ in the PIV branch (at $G^* = 2.0$ ) undergoes transition to the $\phi \approx 0^\circ$ in the PIG branch (at $G^* = 0.5$ ), and then transitions back to the $\phi \approx 180^\circ$ in the PISV branch at the smallest $G^* = 0.1$ . Note that the PIG response here corresponds to the asymptotic increase in $A^*$ , with increasing $U^*$ , as compared with a monotonic increase in $A^*$ for the galloping response of non-axisymmetric cylinders (Abhishek et al. Reference Abhishek, Saha and Sharma2024). Also, about the streamwise centreline of the downstream stationary cylinder, the PIV and PISV branches correspond to symmetric and asymmetric/staggered vibration of the upstream cylinder, respectively.

The above discussion implies that, for larger $U^*$ , the proximity interference leads to the transitions in the branchwise characteristics from the FD branch for the isolated cylinder to PIV, PIG and PISV (with decreasing $G^*$ ). Moreover, as compared with figure 5 for an isolated cylinder, the branches of VIV such as IB, UB and LB become wider at larger gaps $G^* \geqslant 1.0$ while they becomes narrower and eventually disappear as the gap decreases to $G^* = 0.1$ . For various branchwise responses in figure 8, the next section presents a distinct flow physics – in terms of various gap-flow regimes and flow-separation characteristics.

Figure 9.

For the top-most position of the vibrating cylinder, variation of local skin friction $C_f$ along with velocity vectors along the line $L^*$ ( joining the centres of the two cylinders) for (a) isolated cylinder (velocity profile along a horizontal line), (b) PImVIV response at $G^* = 2.0$ , (c) transitional PImVIV–PIG response at $G^* = 1.0$ , (d) PIG response at $G^* = 0.5$ and (e) PISV response at $G^* = 0.1$ . The figure also presents the marked values of angular position $\theta _{f}$ for the front stagnation point and $\theta _{t}/\theta _{b}$ for the separation points at the top/bottom of the vibrating cylinder. Further, a schematic representation of the velocity vector and profile is also presented for the extreme positions of the cylinder.

6.1. The PImVIV response with bi-directional gap-flow (BDGF) regime

For a proximity interference mechanism that leads to the amplitude enhancement for the PImVIV response at larger $G^*$ , figure 10 shows the temporal variation of the vibrational and flow characteristics for a representative PImVIV case at $G^* = 2.0$ and $U^* = 12$ .

For the top-most position of the cylinder, the marked region in figure 10( $b1$ ) shows a gap flow in the upward direction, figure 10( $d1$ ) shows formation of CCW (red) vortex, and figure 10( $c1$ ) shows the associated low-pressure or suction region on the bottom gap side of the cylinder. This dominant suction leads to a negative/downward lift $C_l$ that is anti-phase ( $\phi \sim 180^\circ$ ) with the displacement $y^*$ as seen in figure 10(e). Due to the vortex on the gap side of the vibrating cylinder, figure 10( $a1$ ) shows both top/outer and bottom/gap-side flow separations ( $\theta _{t} = 114^\circ$ and $\theta _{b} = -132^\circ$ ) along with maximum suction or minimum pressure coefficient $C_{p,min}$ almost at the bottom ( $\theta \approx -90^\circ$ ) of the cylinder.

When cylinder moves downward, this downward anti-phase $C_l$ tends to bring the cylinder back to its mean position – called restoring lift force (Mittal & Sharma Reference Mittal and Sharma2022). The restoring lift leads to onset of energy addition to the oscillating cylinder – as seen by increasing positive value of non-dimensional power $\beta$ in figure 10(e) (from the time instant $t^*_1$ to $t^*_2$ ). During the downward motion of the cylinder, figure 10( $c1{-}c2$ ) shows a top-to-bottom shift in the maximum pressure (at the front stagnation point) of the downstream stationary cylinder, hereafter called as proximity pressure. This proximity pressure leads to an increase in the gap-flow rate $Q^*_g$ (figure 10 e) and net circulation of the CCW vortex increases – resulting in a larger $C_{p,min}$ (figure 10 $a2$ ) as compared with that at the top-most position (figure 10 $a1$ ). This enhanced circulation of the anti-phase vortex is responsible for larger magnitudes of $C_l$ and $A^*$ in the PImVIV response as compared with those in the VIV response of an isolated cylinder. However, due to an increase in the downward velocity of the vibrating cylinder, its front stagnation point shifts downward with $\theta _{f} =0^\circ$ to $-4^\circ$ (figure 10 $a2$ ) that contributes to the an upward lift. The combined effect of the downward lift (caused by $C_{p,min}$ on the gap side) and the upward lift (caused by downward shift of front stagnation point) results in an overall decrease in the magnitude of the downward/negative $C_l$ (figure 10 e). This cylinder position at $t^*_2$ instant almost corresponds to the maximum values for suction $C_{p,min}$ and gap-flow rate $Q^*_{g,max} = 0.56$ along with a phase difference $\phi _{g} = 41.3$ (between $Q^*_g$ and $y^*$ ); shown in figure 10(e) slightly before $t^*_2$ .

As the cylinder crosses the mean position, figure 10( $d3$ ) shows that the CCW vortex roll-up and a stronger proximity pressure are seen in figure 10( $c_3$ ). At the same time, figure 10( $b3$ ) shows an increase in the downward gap flow near the trailing edge of the upstream cylinder. This downward gap flow pushes the CCW vortex downward and the vortex re-attaches at the bottom of the downstream cylinder; marked as re-attachment in figure 10( $d4$ ). Further, the increased downward gap flow leads to an increase in strength of the CW vortex (on top of the cylinder) and the minimum suction region starts switching from bottom-to-top side (more clearly seen with a gradual shift in the $C_{p,min}$ in figure 10 $a2{-}a4$ ), leading to an downward-to-upwards transition in the $C_l$ (marked arrow in figure 10 $c3{-}c4$ ) as the cylinder crosses the mean position. For the complete cycle of the periodic vibrations, the region of the gap flow-based suction switching is also marked in figure 10(e) – on the timewise contours plot for pressure coefficient $C_p$ variation on the surface of the cylinder.

As discussed above, the growth of the vortex in the wake of the oscillating cylinder is constraint (within the gap) by the proximity interference from the downstream stationary cylinder. The interaction is referred as wake–cylinder interaction by Alam & Meyer (Reference Alam and Meyer2013) and oscillating wake flow by Mittal & Sharma (Reference Mittal and Sharma2022). Whereas, the vortex is free to grow in the wake for the isolated cylinder. For the PImVIV as compared with VIV response, the bi-directional gap flow (discussed above with figure 9 a,b) increases the circulation of the vortex in the gap, which leads to the enhancement in the restoring lift force and associated amplitude during the PImVIV response. The enhancement of the amplitude of $C_l$ as compared with that of isolated cylinder can be verified from the variation of $C_{l,rms}$ in figure 8( $a3$ ). Further, the excess growth of the vortex in the gap comes at the expense of smaller vortex shedding and oscillation frequency $f^*$ as compared with those during VIV of an isolated cylinder (figure 8 $a2$ ).

6.2. Transitional PImVIV–PIG response with uni-directional non-orthogonal gap-flow (UDnoGF) regime

Similar to figure 10 for the BDGF, figure 11 shows the results for the UDnoGF at the same $U^* = 12$ but a smaller $G^* = 1.0$ . Both the figures are compared below to present a gap-flow-based proximity interference mechanism for the significant enhancement in the amplitude $A^*$ , with the transition from the BDGF to UDGF at $G^* = 1.0$ .

For the top-most position of the vibrating cylinder, as compared with the BDGF in figure 10( $c1$ ), figure 11( $c1$ ) shows a larger proximity pressure on the stationary cylinder that results in the uni-directional flow in the gap. However, the smaller gap $G^* = 1.0$ is large enough to form CCW vortex in the gap but undergoes a slightly early separation on the top/outer side $\theta _{t} = 110^\circ$ and a delayed separation on the bottom/gap side $\theta _{b} = -154^\circ$ (compare figures 10 and 11) – leading to the UDGF being non-orthogonal. Further, on the gap side, a CCW vortex and corresponding minimum suction $C_{p,min}$ are seen (figure 11 $a1{-}d1$ ), leading to an anti-phase restoring lift force in the downward direction, shown in figure 11(e) at instant $t^*_1$ .

As the cylinder moves downward, figure 11( $d2$ ) shows that the vortex in the gap roll-up and moves upwards, figure 11( $a_2$ ) shows an increase in the value to $C_{p,min}$ , and $C_p$ contours in figure 11(e) show a narrowing of the restoring suction region. The narrowing is in contrast to BDGF regime where a similar $C_{p,min}$ increase in figure 10( $a2$ ) along with widening of suction region in figure 10(e) are observed. The narrowing of the suction region is again attributed to the stronger proximity pressure of the downstream cylinder, that restricts the growth of gap-side vortex as shown in figure 11( $d2$ ). Also, figure 11( $a1,a2$ ) shows a reasonably larger downward shift in the front stagnation point $\theta _{f}$ varying from $0^\circ$ to $-12^\circ$ , that contributes to an upward force. However, the downward lift contributed by the gradual shrinking of gap-side suction is not strong enough and the resultant lift becomes nearly zero ( $C_l \sim 0$ ) at time instant $t^*_2$ (figure 11 e) – earlier as compared with the mean position (at $t^*_m$ ) for the BDGF regime (figure 10 e).

The early $C_l = 0$ at $t^*_2$ is followed by a small value of positive/upward lift during the time interval $t^*_2-t^*_m$ , seen in figure 11(e), as the cylinder approaches the mean position. In contrast, after crossing the mean position, a small value of negative/downward lift is seen. Thus, during the entire top-to-bottom motion, the monotonic decrease in $C_l$ leads to the restoring lift force for the BDGF regime (figure 10 e) while the non-monotonic variation in $C_l$ (figure 11 e) leads to a small value of an exciting force (downwards) – near the mean position. The non-monotonic $C_l$ -variation is related to a delayed onset of the gap flow-based switching of restoring suction (from bottom-to-top side of the vibrating cylinder) closer the bottom-most position (shown at $t^*_4$ in figures 11 $c4$ and e), as compared with the that at mean position for BDGF regime (figures 10 $c3$ and e). This delay is attributed to the uni-directional in-phase gap-flow rate $Q^*_g$ with $\phi _g = 4^\circ$ for the UDnoGF regime here as compared with a phase lag $\phi _g = 41^\circ$ in the earlier BDGF regime. Further, across the mean position, a much larger downward shift of the front stagnation point with $\theta _{f}=-12^\circ$ and $-10^\circ$ (figure 11 e) as compared with $-6^\circ$ and $-4^\circ$ (figure 10 e) also contributes to this non-monotonic variation of $C_l$ . The anti-phase ( $\phi = 180^\circ$ ) restoring $C_l$ acts inwards and tends to bring the cylinder back to its mean position, discussed above, while the in-phase ( $\phi = 0^\circ$ ) exciting lift force acts outward and tend to move the cylinder away from the mean position (Mittal & Sharma Reference Mittal and Sharma2022).

In summary, for the UDnoGF and transitional response at $G^* = 1.0$ as compared with the BDGF and PImVIV response at $G^* = 2.0$ , although amplitude of the anti-phase restoring lift is smaller, an in-phase exciting lift also emerges near mean position. This is attributed to the smaller gap-side vortex and an in-phase gap-flow rate ( $\phi _{g} = 4$ ) in the UDnoGF regime (figure 11 e). The smaller the phase difference $\phi _{g}$ leads to the gap flow-based switching of the suction regions more closer to the extreme bottom-most position, which delays the restoring force pulling the cylinder towards the mean position, allowing the cylinder to reach larger amplitude. Thus, for the transitional response at larger $U^*$ , the amplitude enhancement is not attributed to the enhanced gap vortex and lift force (as in BDGF regime) but rather to (a) a smaller gap-side restoring vortex due to stronger proximity pressure and (b) the delay in gap-flow-based switching of restoring suction regions.

6.3. the PIG response with uni-directional orthogonal gap-flow (UDoGF) regime

For a representative case of $G^* = 0.5$ and $U^* = 10$ in the UDoGF regime, at the top-most position of the cylinder, figure 12( $b1,c1$ ) shows that the downstream cylinder is exposed to the free-stream flow that results in a stronger proximity pressure near its front stagnation point. The resulting orthogonal uni-directional gap flow, at the top-most position, leads to a non-separated flow at the bottom/gap side and single separation point on top/outer side with $\theta _{t} = 110^\circ$ , as can be seen in figure 12( $a1,b1$ ). The pressure contours in figures 12( $c1$ ) and 12(e) show that the suction at the gap side is larger than that at the outer side along with a small downward inclination of front stagnation point $(\theta _{f} = -4^\circ$ in figure 12( $a1$ )). The combined effect results in a small dip in $C_l$ , as marked in figure 12(e).

As the velocity of the cylinder increases while moving downwards, the UDoGF leads to an upward inclination of both wake region and separated shear layers (figure 12 $b2{-}b3$ and $d_2{-}d_3$ ). This leads to two separation points on the outer side of the cylinder, marked as $\theta _{t}=100^\circ \textrm { and } 172^\circ$ in figure 12( $a2$ ) and $94^\circ \textrm { and } 172^\circ$ in figure 12( $a3$ ). Also, during the top-to-mean motion of the cylinder, figure 12( $a1{-}a3$ ) shows the one-sided flow separation in the UDoGF regime as compared with the two-sided flow separation in BDGF and UDnoGF regimes. Thus, the orthogonal gap flow close to the vibrating cylinder suppresses the adverse pressure gradient and associated flow separation on the gap side of the cylinder. The suppression and delay of the bottom flow-separation point reveals direct interaction of proximity interference with the boundary layer of the vibrating cylinder, making it the same as the boundary layer–cylinder interaction shown by Alam & Meyer (Reference Alam and Meyer2013). Further, as the upstream cylinder moves closer to the proximity pressure of the downstream cylinder during the time $t^*_2$ to $t^*_3$ , figure 12(e) shows a shrinkage of the gap-side suction and figure 12( $a1{-}a3$ ) shows a downward shift in the front stagnation point ( $\theta _{f} \textrm { varies from } -4^\circ$ to $-20 ^\circ$ ). Their combined effect leads to a local increase in $C_l$ after the initial dip during the top-to-mean motion, shown in figure 12(e).

At the mean position ( $y^*=0$ ), figures 12( $b4$ ) and ( $d4$ ) shows a substantial decrease of both gap flow and the proximity pressure of the downstream cylinder. The $C_p$ contours in figure 12(e) at $t^*_4$ clearly mark the significant dominance of exciting suction at the bottom side as compared with the negligible suction at the top side. This exciting suction increases the negative/downward exciting lift on the cylinder away from the mean position, resulting in energy addition to the cylinder with increasing positive $\beta$ , as shown in figure 12(e). Here, both lift and velocity of the cylinder are in the same downward direction (away from the mean position), which is an attribute of galloping instability (Blevins Reference Blevins1977; Assi & Bearman Reference Assi and Bearman2015). Further, for the mean-to-bottom motion of the cylinder, figure 12( $a5{-}a7$ ) shows the separation point on the bottom half shifts upstream from $\theta _{b} = -138^\circ$ to $-112^\circ$ and front stagnation point shifts towards the centre of front face with $\theta _{f}$ varying from $-16^\circ$ to $2^\circ$ . Finally, for the PIG response, note from figure 12(e) that the gap-flow rate $Q^*_g$ is almost in-phase with $y^*$ ( $\phi _{g} \sim 6^\circ$ ) that results in the gap-flow-based switching of the restoring suction region near the extreme position.

In summary, strong proximity interference affects the boundary layer of the upstream cylinder and generates an uni-directional orthogonal gap flow along with suppression of the adverse pressure gradient, vortex formation on the gap side of the cylinder and associated anti-phase $C_l$ . At the same time, the generation of galloping instability near the mean position results in the in-phase exciting $C_l$ . Whereas, for larger gap $G^*$ in the BDGF regime, the weaker proximity interference interacts primarily with the vortex of the upstream cylinder in the gap, enhancing its circulation. This enhanced circulation of the gap-side vortex leads to enhancements in the anti-phase restoring lift and amplitude, as compared with those during isolated VIV. Further, a detailed mechanism of galloping generation, its absence at larger gaps ( $G^* \geqslant 1.0$ ) and other relevant questions are discussed comprehensively using schematics in upcoming § 7.

6.4. The PISV response with uni-directional one-sided gap-flow (UDosGF)

For a representative case of UDosGF regime with the staggered vibration at $G^* = 0.1$ and $U^* = 14$ , during the top-to-bottom motion, figure 13( $b1{-}b3$ ) shows a slightly unsteady uni-directional gap-flow that is directed upwards, i.e. positive values of gap-flow rate $Q^*$ with very small fluctuation (figure 12 e). Further, figure 13 ( $b1{-}b3$ ) and ( $d1{-}d3$ ) shows that the fluid in the gap forms a small CW vortex in the gap that merges downstream with larger CW vortex from upstream cylinder, and the overall flow in the gap is directed behind the large CCW vortex at the bottom side of the downstream cylinder. This pattern is similar to biased base bleed pattern – observed for two stationary cylinders in staggered arrangement by Lee & Yang (Reference Lee and Yang2009) and Sumner (Reference Sumner2010).

For the entire top-to-bottom motion, as compared with top-to-mean motion in the PIG response, figure 13( $a1{-}a3$ ) shows no separation on the gap side and a slightly varying two separation points ( $\theta _{t} = 114^\circ$ to $116^\circ$ and $178^\circ$ to $180^\circ$ ) on the outer side of the cylinder. The stronger influence of the proximity pressure and uni-directional flow close to upstream cylinder result in significant changes in the boundary layer of the upstream cylinder, suppressing adverse pressure gradients, gap-side flow separation and vortex formation. Thus, figures 13(a) and (c) shows a larger outer-side vortex and a wider outer-side suction with a larger magnitude of $C_{p,min}$ than the gap-side suction on the bottom of the cylinder. Also, there is a slight variation in the front stagnation points with $\theta _{f} = -4$ to $-8^\circ$ . Thus, the pressure distribution around the upstream cylinder always exerts a lift in the upward direction, repelling the cylinder away from the mean position. This repulsive force is evident from figure 13(e), where $C_{l}$ on the upstream cylinder is always large and positive during the entire oscillation cycle.

7.1. Discussion on mechanism of PIG and PISV

In the published literature, various studies are found on the effect of proximity interference from a downstream structure on FIV of an elastically mounted cylinder. However, for this problem, there is a scope to present a more comprehensive flow mechanism as compared with that reported in the literature (Bokaian & Geoola, Reference Bokaian and Geoola1984a ; Mittal & Sharma Reference Mittal and Sharma2022; Su et al. Reference Su, He, Zhang and Feng2022). Thus, for the flow mechanisms that lead to (a) amplitude enhancement in the PImVIV response, (b) generation of galloping instability in the PIG response and (c) onset of PISV response, we attempt to answer the four questions as follows:

1. (i) What is the mechanism of galloping instability generation during the PIG response, and how it is different from the galloping of an isolated non-axisymmetric cylinder?

2. (ii) Why does the proximity interference result in maximum amplitude of vibration at an intermediate gap $G^*$ ?

3. (iii) Why does the amplitude response is asymptotically increasing in PIG as compared with the monotonic increase in galloping of an isolated non-axisymmetric cylinder?

4. (iv) Why does the proximity interference lead to the onset of staggered vibration (PISV) at a critical reduced velocity $U^*_c$ ?

For each of the four question, the mechanisms are presented in the separate subsection below. The mechanisms are presented with the help of a schematic representation of flow phenomena (during the top-to-bottom motion of the cylinder) for the BDGF regime in PImVIV response and the UDoGF regime in the PIG response in figures 16(a) and 16(b), respectively. Note that, in the forthcoming discussion, a stabilising force acts opposite to the direction of motion ( $\beta \lt 0$ ) and decelerates the cylinder while a destabilising force acts in the same direction ( $\beta \gt 0$ ) and accelerates the cylinder.

Figure 16.

Schematic representation of the flow dynamics during top-to-bottom motion of the cylinder for ( $a_1{-}a_3$ ) the BDGF regime in PImVIV response and ( $b_1{-}b_3$ ) the UDoGF regime in PIG response. Panels (c) and (d) present the temporal variation of displacement $y^*$ , velocity $\dot {y^*}$ , and lift coefficient $C_l$ . For (c) and (d), note that the time instant (1), (2) and (3) correspond to the schematic of PIG response in ( $b_1$ ), ( $b_2$ ) and ( $b_3$ ), respectively. For the schematics in (a) and (b), the coloured regions represent high/low pressure – with red and blue/cyan corresponding to high and low pressure at the front and separated regions of the cylinder, respectively. Further, the blue as compared with cyan corresponds to a larger or dominant suction region.