3.1.1 Stationary cylinder with a flexible splitter plate ( ${\mathcal{C}}_{stat}{\mathcal{P}}_{vib}$ )

Figure 2.

Free vibration of a flexible splitter plate attached to a fixed cylinder ( ${\mathcal{C}}_{stat}{\mathcal{P}}_{vib}$ ): variation of non-dimensionalized natural frequencies, $F_{np}^{k}$ , with reduced velocity, $U_{p}^{\ast }$ , for the first four modes of vibration. The corresponding mode shapes of the plate are shown for $L=3.5D$ .

The natural frequency of the flexible splitter plate is estimated for defining $U_{p}^{\ast }$ . Each plate, of certain flexibility, is associated with infinite natural frequencies and eigenmodes. It is modelled as a cantilever beam and the first few modes are estimated using the Euler–Bernoulli beam theory. We utilize the natural frequency of the first mode of the plate, thus obtained, to characterize its flexibility. To this extent, the reduced velocity of the plate is defined as $U_{p}^{\ast }=1/F_{np}^{1}=(6.191(L/D)^{2}/(h/D))\sqrt{m^{\ast }/Ae}$ . The details of the estimation are in appendix A. We note that $U_{p}^{\ast }=0$ corresponds to a rigid plate. The value of $U_{p}^{\ast }$ increases with an increase in flexibility of the plate. Figure 2 shows the variation of $F_{np}^{k}$ with $U_{p}^{\ast }$ for the first four eigenmodes. The natural frequency of vibration for each mode varies inversely with variation in $U_{p}^{\ast }$ . The characteristic mode shape associated with each of the first three natural frequencies is also shown in the figure.

Figure 3.

Free vibration of an elastically supported circular cylinder with flexible plate ( ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ ): variation of non-dimensionalized natural frequency, $F_{nsp}^{k}$ , with the reduced velocity, $U_{p}^{\ast }$ , for various values of $U_{s}^{\ast }$ . The natural frequencies associated with different eigenmodes lie in non-overlapping regions, shown with different shades. These regions are bounded by the $F_{np}^{k}$ versus $U_{p}^{\ast }$ curves, drawn with solid lines, which are also shown in figure 2.

3.1.2 Elastically supported cylinder with a flexible splitter plate ( ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ )

The system can undergo combined heave oscillations of spring–mass and bending vibrations of the plate. The compliance of the splitter plate and elastic support is characterized by the reduced speeds $U_{p}^{\ast }$ and $U_{s}^{\ast }$ , respectively. Figure 3 shows the variation of the first five natural frequencies of the system ( $F_{nsp}^{k}$ , $k=1,2,\ldots ,5$ ) with $U_{p}^{\ast }$ and $U_{s}^{\ast }$ . We note that the system ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ degenerates to ${\mathcal{C}}_{stat}{\mathcal{P}}_{vib}$ , i.e. a rigid support, for $U_{s}^{\ast }=0$ . Consequently, the curve corresponding to $F_{nsp}^{k}$ , shown in solid line for $U_{s}^{\ast }=0$ in figure 3, is identical to the curve for $F_{np}^{k}$ in figure 2 for all $k$ . It is seen from figure 3 that each mode occupies a distinct region in the $F_{nsp}^{k}$ – $U_{p}^{\ast }$ plane. It can be shown (see appendix A) that each region can be collapsed into a curve by using the similarity variables $U_{s}^{\ast }F_{nsp}^{k}$ and $U_{p}^{\ast }F_{nsp}^{k}$ . This is shown in figure 4 along with mode shapes for a few points on these curves. The figure is useful in understanding as to how the addition of the heave degree of freedom to ${\mathcal{C}}_{stat}{\mathcal{P}}_{vib}$ , in terms of the elastic support, modifies the eigenmodes/values of the system ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ . We make the following four observations. (i) The cylinder and plate tip move in phase for the odd-numbered modes and out of phase for the even-numbered ones. (ii) The first mode of ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ is similar to that for ${\mathcal{C}}_{stat}{\mathcal{P}}_{vib}$ . (iii) For $k>1$ , $F_{nsp}^{k}$ as well as the shape of the plate in the $k\text{th}$ eigenmode of ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ are, respectively, close to $F_{np}^{k-1}$ and the $(k-1)\text{th}$ eigenmode of ${\mathcal{C}}_{stat}{\mathcal{P}}_{vib}$ for large values of $U_{s}^{\ast }$ and fixed $U_{p}^{\ast }$ . For small $U_{s}^{\ast }$ , they are close to the $F_{np}^{k}$ and the $k\text{th}$ eigenmode of ${\mathcal{C}}_{stat}{\mathcal{P}}_{vib}$ . (iv) As a consequence of observation (iii), we see from figure 4 that both $F_{nsp}^{k}$ and $F_{nsp}^{k+1}$ can be close to $F_{np}^{k}$ , for suitable values of $U_{p}^{\ast }$ and $U_{s}^{\ast }$ . In either case, the shape of the deformed plate is similar to that in the $k\text{th}$ mode of ${\mathcal{C}}_{stat}{\mathcal{P}}_{vib}$ . However, the plate tip is in phase with the cylinder in one case and out of phase in the other.

Figure 4.

Free vibration of an elastically supported circular cylinder with flexible splitter plate ( ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ ): figure 3 that shows variation of natural frequency with $U_{p}^{\ast }$ and $U_{s}^{\ast }$ is re-plotted in terms of similarity variables $U_{s}^{\ast }F_{nsp}^{k}$ and $U_{p}^{\ast }F_{nsp}^{k}$ . Curves I, II, III and IV, shown in different colours, correspond to first four natural frequencies. Also shown are the different mode shapes for $L=3.5D$ .

3.6.1 $U_{s}^{\ast }=6$ , effect of flexibility of splitter plate ( $U_{p}^{\ast }$ ) on VIV

Figure 19.

The $Re=150$ , $U_{s}^{\ast }=6$ , $L=3.5D$ flow past a spring-mounted circular cylinder with a flexible splitter plate ( ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ ): time histories of cylinder location and that of the plate tip for (a) $U_{p}^{\ast }=5.36$ (regime-i) and (b) $U_{p}^{\ast }=12.8$ (regime-ii). Vorticity field at a time instant when the cylinder displacement is maximum, and marked in the time histories, is shown below the respective time histories. For reference, the undeformed structure is also shown.

Figure 20.

The $Re=150$ , $U_{s}^{\ast }=6$ , $U_{p}^{\ast }=5.68$ , $L=3.5D$ flow past a spring-mounted circular cylinder with a flexible splitter plate ( ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ ): (a) time histories of cylinder and plate-tip displacements and (b) vorticity field showing $P+S$ mode of vortex shedding. For comparison, the $2S$ mode of vortex shedding for $U_{p}^{\ast }=12.8$ is shown in (c).

Figure 18 shows the variation of the normalized amplitude of vibration of the cylinder as well as the tip of the splitter plate, relative to the centre of the cylinder, with $U_{p}^{\ast }$ . Also shown is the phase between the vibration of cylinder and tip of the splitter plate as well as the vibration frequency, normalized with the first and second natural frequency of the structure. Two regimes are identified from this figure. Regime-i: $U_{p}^{\ast }<5.68$ , wherein the system response is closer to the first natural mode of vibration; and regime-ii: $U_{p}^{\ast }>5.68$ , where the response is closer to the second mode. The cylinder vibration amplitude is lowest at $U_{p}^{\ast }=5.68$ . The response amplitude of the splitter plate, relative to the cylinder, increases with an increase in $U_{p}^{\ast }$ . Figure 19 shows the time histories of the response of the structure as well as the unsteady flow for two values of $U_{p}^{\ast }$ ; one each in the two regimes. As seen from the figure, the motions of the cylinder and the tip of the splitter plate are in phase for $U_{p}^{\ast }=5.36$ (regime-i) and almost out of phase for $U_{p}^{\ast }=12.8$ (regime-ii). The deformed shape of the plate confirms the dominance of the first mode in regime-i and that of the second mode in regime-ii. We recall that, for fixed $U_{s}^{\ast }$ , the first mode is associated with a lower natural frequency compared to the second mode (figure 3). This difference is also seen in the vortex shedding frequency in the flow pictures for the two values of $U_{p}^{\ast }$ . However, the mode of vortex shedding in both regimes is 2S, wherein two vortices of opposite sign are shed during one cycle of vortex shedding. Figure 20(c) shows the flow at various time instants during one cycle of plate motion for $U_{p}^{\ast }=12.8$ . The generation of vortices and their convection in the wake are demonstrated in the figure.

The case of $U_{p}^{\ast }=5.68$ is particularly interesting; it represents the transition between regime-i and regime-ii. The time histories of the response of the structure are shown in figure 20(a). Fast Fourier transform of the signals reveals that there are two frequencies at play: their non-dimensional values are $0.106$ and $0.209$ . The beats in the time histories are a consequence of the larger frequency being slightly less than twice the smaller one. The lower frequency is closer to the variation observed along the branch corresponding to regime-i, while the larger one belongs to the branch for regime-ii. The phase difference between the motion of cylinder and plate tip, corresponding to each of the frequency variations, was estimated. It is found that the motion of cylinder and plate tip are in phase for the lower frequency and out of phase for the larger frequency, again consistent with the characteristics of regime-i and regime-ii. The phase and normalized frequencies are marked with filled symbols in figure 18 to indicate that they exhibit multiple values during the transition. The flow at various time instants during one cycle of plate motion is shown in figure 20(b). For comparison, the flow for $U_{p}^{\ast }=12.8$ is shown as well in figure 20(c). Unlike the 2S mode of shedding observed in regime-i and regime-ii (figure 19 a,b), the vortex shedding pattern for this transitional state ( $U_{p}=5.68$ ) is $P+S$ (figure 20 b). A single vortex and a pair of counter-rotating vortices are formed during each cycle of vortex shedding.

3.6.2 $U_{s}^{\ast }=22$ , effect of flexibility of splitter plate ( $U_{p}^{\ast }$ ) on galloping

Figure 21.

The $Re=150$ , $U_{s}^{\ast }=22$ , $L=3.5D$ flow past a spring-mounted circular cylinder with a flexible splitter plate ( ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ ): variation of (a) normalized amplitude of cylinder oscillation $A_{c}^{\ast }$ and plate-tip oscillation $A_{p}^{\ast }$ , (b) phase difference between the motion of cylinder and plate tip $\unicode[STIX]{x1D6F7}$ and (c) frequency ratios $f_{sp1}^{\ast }$ and $f_{sp2}^{\ast }$ calculated, respectively, using the first and second natural frequencies of the ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ modes with plate flexibility $U_{p}^{\ast }$ .

Figure 22.

The $Re=150$ , $U_{s}^{\ast }=22$ , $L=3.5D$ flow past a spring-mounted circular cylinder with a flexible splitter plate ( ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ ): time histories of cylinder and plate tip positions for (a) $U_{p}^{\ast }=4.68$ (galloping) and (b) $U_{p}^{\ast }=7.4$ (VIV). Vorticity field at a time instant when the cylinder displacement is maximum is shown below the respective time histories. For reference, the undeformed structure is also shown.

The effect of flexibility of the plate on galloping is explored. A reasonably large value of $U_{s}^{\ast }$ is chosen for the study so that the galloping response, if present, is of relatively large amplitude. Figure 21 shows the variation of the normalized amplitude of vibration of the cylinder as well as the tip of the splitter plate, relative to the centre of the cylinder, with $U_{p}^{\ast }$ . Also shown is the phase between the vibration of cylinder and tip of the splitter plate as well as the vibration frequency, normalized with the first and second natural frequency of the structure. Two regimes are identified from this figure. Regime-i: $U_{p}^{\ast }\leqslant 4.68$ , wherein the system exhibits galloping response; and regime-ii: $U_{p}^{\ast }>4.68$ , where the system undergoes VIV. Interestingly, the phase between the motion of the cylinder and plate tip also undergoes a jump at $U_{p}^{\ast }=4.68$ ; the motion is in phase in regime-i and nearly out of phase in regime-ii. Figure 22 shows the time histories of the response at a representative $U_{p}^{\ast }$ ( $=4.68$ ) in the galloping regime (regime-i). In this regime, the motions of the cylinder and tip of the plate are in phase. In fact, the tip of the plate moves very little relative to the cylinder. It can be seen from the figure that the mean position of the cylinder and plate tip is different from that in the undeformed configuration. A similar behaviour was also observed by Wu et al. (Reference Wu, Qiu, Shu and Zhao2014a ) for certain cases of an elastically mounted cylinder with an attached filament. Also shown is the vorticity field for the instant when the cylinder and the tip of the plate are at their maximum displacement. The vortex shedding occurs a few plate lengths downstream in the wake and with a relatively low frequency. As seen from figure 21, the amplitude of the cylinder response undergoes a sharp reduction when $U_{p}^{\ast }$ is increased beyond $4.68$ and the system enters the regime of VIV (regime-ii). The amplitude of the response of the tip of the plate rises sharply. It increases on increasing $U_{p}^{\ast }$ up to approximately $5.0$ . Beyond $U_{p}^{\ast }=5$ , the response amplitude decreases with further increase in $U_{p}^{\ast }$ . The system vibrates at a frequency close to the natural frequency of the second mode. The time histories of the response are shown in figure 22(b) for a representative $U_{p}^{\ast }$ ( $=7.4$ ) in this regime (regime-ii). Also shown is the flow when the cylinder is at the peak displacement. The vortex shedding, in this case, occurs in the very near wake of the cylinder. In both the flow pictures (regime-i and regime-ii), the vortex shedding is of the type 2S, where two vortices of opposite signs are shed during each cycle of shedding.

The studies for $U_{s}^{\ast }=6$ and $22$ can be utilized to identify optimal flexibility of the plate to minimize FIV. Large-amplitude galloping response can be avoided by choosing a splitter plate with $U_{p}^{\ast }\geqslant 4.68$ . In fact, for $U_{s}^{\ast }=22$ , the amplitude of galloping response decreases with an increase in $U_{p}^{\ast }$ . Therefore, the larger the value of $U_{p}^{\ast }$ , the smaller is the amplitude of galloping. However, large values of $U_{p}^{\ast }$ are associated with large amplitude of VIV. The amplitude of VIV, for $U_{s}^{\ast }=6$ , is minimum for $U_{p}^{\ast }=5.68$ . Therefore, $U_{p}^{\ast }=5.68$ corresponds to the optimal flexibility of the plate for restricting both galloping and VIV.

3.6.3 Effect of length of splitter plate on VIV ( $U_{s}^{\ast }=6$ ) and galloping ( $U_{s}^{\ast }=22$ )

Figure 23.

The $Re=150$ , $U_{s}^{\ast }=6$ , $L=1.5D$ flow past a spring-mounted circular cylinder with a flexible splitter plate ( ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ ): variation of (a) normalized amplitude of cylinder oscillation $A_{c}^{\ast }$ and plate-tip oscillation $A_{p}^{\ast }$ , (b) phase difference between the motion of cylinder and plate tip $\unicode[STIX]{x1D6F7}$ and (c) frequency ratios $f_{sp1}^{\ast }$ and $f_{sp2}^{\ast }$ calculated, respectively, using the first and second natural frequencies of the ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ modes with plate flexibility $U_{p}^{\ast }$ .

Figure 24.

The $Re=150$ , $U_{s}^{\ast }=22$ , $L=1.5D$ flow past a spring-mounted circular cylinder with a flexible splitter plate ( ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ ): variation of (a) normalized amplitude of cylinder oscillation $A_{c}^{\ast }$ and plate-tip oscillation $A_{p}^{\ast }$ , (b) phase difference between the motion of cylinder and plate tip $\unicode[STIX]{x1D6F7}$ and (c) frequency ratios $f_{sp1}^{\ast }$ , $f_{sp2}^{\ast }$ and $f_{sp3}^{\ast }$ calculated, respectively, using the first, second and third natural frequencies of the ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ modes with plate flexibility $U_{p}^{\ast }$ .

To investigate the effect of plate length on FIV, we study the response of the ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ system with $L=1.5D$ and compare it to that for $L=3.5D$ . Figures 23 and 24 present the variation of normalized amplitudes, the phase difference $\unicode[STIX]{x1D6F7}$ and the frequency ratios for $U_{s}^{\ast }=6$ and $22$ , respectively, just as figures 18 and 21 do for $L=3.5D$ . Figure 23(b,c) shows that for $L=1.5D$ and $U_{p}^{\ast }\lesssim 2.6$ , the cylinder and the plate tip oscillate nearly in phase with a frequency close to $F_{nsp1}$ . On the other hand, the two oscillate out of phase and with a frequency close to $F_{nsp2}$ for $U_{p}^{\ast }\gtrsim 8.9$ . Thus, as in the case of $L=3.5D$ , VIV response of the system is dominated by the first ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ mode at low $U_{p}^{\ast }$ and the second mode at large $U_{p}^{\ast }$ . However, unlike for the longer plate, the transition from the first mode behaviour to the second is gradual. Figure 23(a) shows that the variation in cylinder and plate-tip amplitudes is also qualitatively similar to that for $L=3.5D$ (figure 18 a). However, amplitude at each value of $U_{p}^{\ast }$ is smaller compared to the corresponding value for $L=3.5D$ . This is primarily because, for a given $U_{p}^{\ast }$ , the $U_{s}^{\ast }$ , at which the amplitude of plate vibration is maximum, decreases with a decrease in $L$ (figure 14 a). The amplitude of plate vibration is close to maximum at $U_{s}^{\ast }\sim 6$ for $L=3.5D$ while it occurs at $U_{s}^{\ast }\sim 4$ for $L=1.5D$ .

Figure 24 shows that, as for $L=3.5D$ (figure 21), the critical value of $U_{p}^{\ast }$ , for onset of galloping, is $4.68$ . For both the plate lengths, galloping occurs for $U_{p}^{\ast }\leqslant 4.68$ (regime-i). However, the frequency of vibration is closer to $F_{nsp1}$ for the shorter plate, compared to the longer one. Also, the amplitude of galloping is higher for $L=1.5D$ . The vortex shedding is relatively strong, in the near wake, for the shorter plate even during galloping. Such a phenomenon has also been observed in the past and is referred to as VIV–galloping interaction (Liang et al. Reference Liang, Wang and Hu2018). Regime-ii ( $U_{p}^{\ast }>4.68$ ), for $L=1.5D$ , can be further divided into two subregimes: regime-ii(a) and regime-ii(b). Regime-ii(a), $4.68<U_{p}^{\ast }\leqslant 7.4$ , is similar to the regime-ii for $L=3.5D$ (figure 21). No galloping is observed in this regime, leading to a drastic reduction in the amplitude of cylinder vibration. However, the amplitude of plate vibration increases sharply as the system transits from regime-i to regime-ii(a). The cylinder and plate oscillate almost out of phase, with a frequency close to the second natural frequency $F_{nsp2}$ . In regime-ii(b), which occurs for $U_{p}^{\ast }\gtrsim 7.4$ , the cylinder and the plate tip again oscillate in phase. Both are associated with a relatively small amplitude. Figures 3 and 4 show that the plate and cylinder vibrate in phase for the first and third modes of vibration of ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ , while they are out of phase for the second mode. Therefore, either the first or third mode (or both) is significant in regime-ii(b). Figure 24(c) shows that the frequency of vibration in this regime is very different from the first natural frequency ( $f_{sp1}^{\ast }\gg 1$ ). It also shows an increase in $f_{sp3}^{\ast }$ , towards a value of $1$ , with an increase in $U_{p}^{\ast }$ . This indicates that the third mode of vibration, and not the first one, is most significant during regime-ii(b). Figure 23(a) shows that the smallest amplitude of vibration is observed at $U_{p}^{\ast }=7.4$ . Since galloping does not occur for $U_{p}^{\ast }>4.68$ , $U_{p}^{\ast }=7.4$ is close to optimal flexibility of the plate, for $L=1.5D$ , for minimizing flow-induced oscillation of the cylinder. We recall that the optimal flexibility, for the longer plate with $L=3.5D$ , is lower and corresponds to $U_{p}^{\ast }=5.68$ .

3.6.4 Effect of mass ratio and damping on the FIV response

Figure 25.

The $Re=150$ , $U_{p}^{\ast }=7.4$ , $L=3.5D$ flow past a spring-mounted circular cylinder with a flexible splitter plate ( ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ ) for three different $m^{\ast }$ : variation of the normalized amplitude of (a) cylinder oscillation $A_{c}^{\ast }$ and (b) plate-tip vibration $A_{p}^{\ast }$ with reduced velocity $U_{s}^{\ast }$ . Shown in (c) is the phase difference, $\unicode[STIX]{x1D6F7}$ , between the vibration of cylinder and plate-tip displacements.

Figure 26.

The $Re=150$ , $U_{p}^{\ast }=7.4$ , $L=3.5D$ flow past a spring-mounted circular cylinder with a flexible splitter plate ( ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ ) for three different $m^{\ast }$ : variation of frequency ratio calculated using the natural frequency of the (a) first ( $f_{sp1}^{\ast }$ ) and (b) second ( $f_{sp2}^{\ast }$ ) mode of the system ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ with reduced velocity $U_{s}^{\ast }$ .

The effect of mass ratio on the FIV of the ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ system is studied via computations with three values of mass ratio, $m^{\ast }=5,10$ and $50$ . Values of $U_{p}^{\ast }$ and $L$ for the study are fixed at $7.4$ and $3.5D$ while $U_{s}^{\ast }$ is varied. We recall that $(U_{p}^{\ast },L)=(7.4,3.5D)$ is associated with a smaller amplitude of cylinder vibration, compared to that of an isolated cylinder, during VIV and, unlike a cylinder with rigid splitter plate, does not gallop at large values of $U_{s}^{\ast }$ (see figure 16). Figure 25 shows the variation, with $U_{s}^{\ast }$ , of amplitude of oscillation of the cylinder and the plate tip as well as the phase difference between the two for different $m^{\ast }$ . The variation of frequency ratios, $f_{sp1}^{\ast }$ and $f_{sp2}^{\ast }$ , is shown in figure 26. The response is qualitatively quite similar for all $m^{\ast }$ . Figure 25(c) shows that the lower values of $U_{s}^{\ast }$ are associated with nearly in-phase motion of the cylinder and the plate. Further, it is seen from figure 26(a) that the ratio $f_{sp1}^{\ast }$ is near unity for these values of $U_{s}^{\ast }$ . It is closer to unity for larger $m^{\ast }$ . As also discussed earlier, these are characteristics of a response dominated by the first ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ mode. Similar conclusions can be drawn, from figures 25(c) and 26(b), about the role of the second ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ mode in the higher $U_{s}^{\ast }$ range. Figure 25 shows that the transition from mode 1 to 2 is associated with a sharp jump in the amplitude of vibration of the cylinder as well as the plate. Although the nature of this transition is the same across all values of $m^{\ast }$ considered, the quantum of jump and the $U_{s}^{\ast }$ at which it occurs depends significantly on $m^{\ast }$ . With an increase in $m^{\ast }$ , it occurs at larger $U_{s}^{\ast }$ and the magnitude of the jump decreases. Interestingly, the vibration amplitude achieves a maximum in mode 2, immediately after the jump. Such a delay in maximum amplitude, with respect to $U_{s}^{\ast }$ , was also observed recently by Sahu et al. (Reference Sahu, Furquan, Jaiswal and Mittal2019) for transverse free vibration of a cylinder with a rigid plate. However, unlike in the present study, the cylinder with a rigid plate is associated with only one mode of vibration. Therefore, in their study, a change of mode is not possible and no jump in amplitude of response is observed. Irrespective of $m^{\ast }$ , $2S$ mode of vortex shedding is observed for all $U_{s}^{\ast }$ .

Figure 27.

The $Re=150$ , $m^{\ast }=10$ , $U_{p}^{\ast }=7.4$ , $L=3.5D$ flow past a spring-mounted circular cylinder with a flexible splitter plate ( ${\mathcal{C}}_{vib}{\mathcal{P}}_{vib}$ ) for three different $\unicode[STIX]{x1D701}$ : variation of the normalized amplitude of (a) cylinder oscillation $A_{c}^{\ast }$ and (b) plate-tip vibration $A_{p}^{\ast }$ with reduced velocity $U_{s}^{\ast }$ .

The effect of damping ratio, $\unicode[STIX]{x1D701}$ , on the FIV response is studied via three values of $\unicode[STIX]{x1D701}$ (0, 0.001 and 0.01). We note that a number of experimental and numerical studies of VIV of an isolated cylinder have used low values of $\unicode[STIX]{x1D701}$ of the order of $10^{-3}$ (Bishop & Hassan Reference Bishop and Hassan1964; Anagnostopoulos & Bearman Reference Anagnostopoulos and Bearman1992; Brika & Laneville Reference Brika and Laneville1993; Khalak & Williamson Reference Khalak and Williamson1999; Triantafyllou et al. Reference Triantafyllou, Hover, Techet, Yue, Benaroya and Wei2003; Klamo, Leonard & Roshko Reference Klamo, Leonard and Roshko2006; Blevins & Coughran Reference Blevins and Coughran2009; Prasanth et al. Reference Prasanth, Premchandran and Mittal2011; Law & Jaiman Reference Law and Jaiman2017). Computations are carried out for each $\unicode[STIX]{x1D701}$ at $U_{p}^{\ast }=7.4$ , $m^{\ast }=10$ and $L=3.5D$ and varying $U_{s}^{\ast }$ . Figure 27 shows the variation of amplitude of vibration of cylinder and plate tip with $U_{p}^{\ast }$ for all three cases. The responses for different values of $\unicode[STIX]{x1D701}$ are quite close. The only discernible difference in the response is a slight reduction in the peak value of the amplitude with increasing $\unicode[STIX]{x1D701}$ . This is consistent with the earlier studies of VIV of an isolated cylinder at low $Re$ (Bahmani & Akbari Reference Bahmani and Akbari2010; Prasanth et al. Reference Prasanth, Premchandran and Mittal2011).