4.1. Vibration responses

A cylinder mounted on air bearings without springs can be considered inelastic, with zero stiffness and zero natural frequency. Here, we present results for the inelastic cylinder placed in the wake of a fixed cylinder (figure 1), where the free-stream flow velocity is changed from 0.098 m s⁻¹ to 0.59 m s⁻¹. Since the cylinder has a zero natural frequency, the free-stream flow velocity cannot be normalised to U _r but to Re. Figure 3(a–c) shows dependencies of A/D on Re ( = 1.97 × 10³–1.18 × 10⁴) and L/D ( = 2.0–6.0). The cylinder undergoes large-amplitude vibrations once Re exceeds a critical value, depending on L/D. At L/D = 2.0, 2.5 and 3.0, violent vibrations prevail at Re > 8.3 × 10³, 6.7 × 10³ and 6.7 × 10³, respectively (figure 3 a). Here the small-amplitude vibrations (Re < 8.3 × 10³, 6.7 × 10³ and 6.7 × 10³ for L/D = 2.0, 2.5, and 3.0, respectively) are defined as the DB while the large-amplitude vibrations (Re > 8.3 × 10³, 6.7 × 10³ and 6.7 × 10³ for L/D = 2.0, 2.5 and 3.0, respectively) are defined as the galloping branch (GB). The reason behind why they are called so is provided later. The onset of the violent vibrations is characterised by jumps in A/D from A/D = 0.05 to 0.7–0.9 depending on L/D. Similar observations are made for L/D = 3.5, 4.0 and 4.5, albeit with some distinct features (figure 3 b). First, a small peak is observed at Re = 3.6 × 10³ in the DB, slightly exceeding the peak value observed for L/D ≤ 3.0 in figure 3(a). Second, the Re values at the onsets of vibrations are smaller than those for L/D ≤ 3.0. The A/D jumps from A/D = 0.05 to 0.5–1.2 depending on L/D. The value of A/D in the GB levels off earlier for L/D = 2.0 and 3.0 than for L/D = 3.5 and 4.0. A further increase in Re for L/D = 2.5 and 4.5 may curtail A/D to become levelled off. Alam (Reference Alam2014) for two fixed tandem cylinders found that fluctuating lift forces are the largest for L/D ≈ 2.5 in the reattachment regime and for L/D = 3.5–4.5 in the coshedding regime, which may be a reason why the A/D for these L/D values is yet to level off. In addition, when A/D is sufficiently large, exceeding a certain limit depending on L/D, the cylinder reaches amplitude outside the wake-interference region where the quasi-steady lift force declines with increasing cylinder displacement (Assi et al. Reference Assi, Bearman, Carmo, Meneghini, Sherwin and Willden2013; Alam & Mayer Reference Alam and Meyer2013). Assi et al. (Reference Assi, Bearman, Carmo, Meneghini, Sherwin and Willden2013) in their study of tandem fixed cylinders with L/D = 4.0 and Re = 9.6 × 10³–1.92 × 10⁴ showed that the time-mean lift coefficient, which consistently acts toward the wake centreline of the upstream cylinder, increases linearly with increasing lateral separation y _o /D of the downstream cylinder from y _o /D = −1 to 1 while its magnitude decreases nonlinearly with increasing |y _o /D| > 1. The lift force behaves like a linear spring for −1 < y _o /D < 1. In contrast, for L/D = 2.0, the time-mean lift coefficient becomes a linear function of y _o /D only within a narrow range of −0.4 < y _o /D < 0.4 (Alam & Meyer Reference Alam and Meyer2013). This explains why A/D levels off at a smaller value of A/D for L/D = 2.0 compared with L/D = 4.0. In the vibration cases, plateaued A/D reaches higher values than the corresponding y _o /D values observed in fixed cylinder cases due to the dynamic response of the vibrating cylinder. For L/D = 2.5 and 4.5, the present Re range is insufficient to capture the levelling-off behaviour. However, based on analogy with the other spacing ratios, we anticipate that the cylinder at these spacings also would reach a maximum amplitude at higher Re before levelling off.

Figure 3.

(a–c) Dependence of vibration amplitude A/D on Re and L/D. (d) Isocontours of A/D corresponding to (a–c). The white dashed line in (d) represents the boundary between GB and DB while the dashed-dotted line distinguishes RF and CF.

The A/D dependencies on L/D = 5.0–6.0 are qualitatively similar (figure 3 c) but different from those on L/D ≤ 4.5 (figure 3 a,b). The jump in A/D becomes faint, and the onsets of GB prevail at small Re. The A/D in the GB generally increases with Re but decreases with increasing L/D from 5.0 to 6.0. Overall, the A/D is largest at L/D = 2.5 in the reattachment flow regime and at L/D = 3.5 in the coshedding flow regime where A/D declines with increasing L/D from 3.5 to 6.0. Although L/D = 3.5 – 4.5 (figure 3 b) and L/D = 5.0 – 6.0 (figure 3 c) both fall in the coshedding flow regime, why the vibration responses differ between the two L/D regimes requires an explanation. It has been well documented that two tandem fixed cylinders undergo higher fluctuating lift forces at L/D = 3.5–4.5 with nearly in-phase vortex sheddings from the two cylinders than at L/D = 5.0–6.0 with nearly antiphase vortex sheddings (Alam Reference Alam2016). That is, the vortex shedding phase and associated fluctuating lift forces dictate the vibration response.

Assi et al. (Reference Assi, Bearman, Carmo, Meneghini, Sherwin and Willden2013) found that, for L/D = 4.0–10.0, A/D increases almost linearly with increasing Re. The rate of increase in A/D, however, diminishes as L/D increases. Consequently, for a given Re, the value of A/D decreases with increasing L/D. When the cylinder was given elasticity (mounted on springs) with m* $\zeta$ _air = 0.018, they observed wake-induced vibrations following VIV where A/D increases with U _r for L/D = 4.0–6.0, remains more or less constant for L/D = 8.0 and decreases for L/D = 10.0 and 20.0.

The A/D isocontours provide a dashing picture of the A/D dependence on L/D and Re (figure 3 d). The dashed line divides the entire regime into DB and GB, while the dashed-dotted line distinguishes reattachment flow (RF) and coshedding flow (CF). This results in four regimes: (i) RF, DB; (ii) RF, GB; (iii) CF, DB; and (iv) CF, GB. Interestingly, A/D reaches its maximum in the GB at L/D = 3.5, corresponding to CF. This L/D value is adjacent to the border between RF and CF, indicating an increase in A/D during the transition from RF to CF. Furthermore, for fixed cylinders, the fluctuating lift force on the downstream cylinder also intensifies immediately following the transition (Alam, Sakamoto & Zhou Reference Alam, Sakamoto and Zhou2006; Kim et al. Reference Kim, Alam, Sakamoto and Zhou2009). The value of A/D becomes smaller when L/D increases or decreases from 3.5 for a given Re. The isocontours are more densely distributed at L/D > 3.0, suggesting a relatively stronger dependence of A/D on L/D in the CF regime.

Given that a positive stiffness to the cylinder is required to generate vibration, the Re increase may enhance the flow-induced stiffness, and the cylinder gaining sufficient stiffness suffers violent vibrations. In addition, a higher L/D leads to a greater increase in the stiffness with Re (shown later), which explains why vibration commences at a smaller Re for a larger L/D. Assi et al. (Reference Assi, Bearman, Carmo, Meneghini, Sherwin and Willden2013) at m* = 2.6, L/D = 4 –10 observed decreased A/D with L/D but increased A/D with Re for the range of Re = 2 × 10³–2.2 × 10⁴ examined. The A/D, even at the smallest Re = 2 × 10³ examined, was greater than 0.2 in Assi et al. (Reference Assi, Bearman, Carmo, Meneghini, Sherwin and Willden2013). In other words, they did not observe any DB but GB. The difference could be attributed to a very low m* = 2.6 in Assi et al. (Reference Assi, Bearman, Carmo, Meneghini, Sherwin and Willden2013) compared with m* = 11.1 in the present experiment.

4.2. Effective natural frequency, effective stiffness and vortex shedding frequency

Here, the cylinder oscillation frequency is referred to as the effective natural frequency f $_{\!w}$ of the cylinder undergoing the wake-induced stiffness (effective stiffness) as it is proved later that the oscillation frequency essentially equals the effective natural frequency. Here, the effective stiffness refers to the total stiffness, which encompasses both the stiffness resulting from the cylinder offset (Assi et al. Reference Assi, Bearman, Carmo, Meneghini, Sherwin and Willden2013) and the stiffness arising from the cylinder motion. The values of f $_{\!v}$ and f $_{\!w}$ , identified from the power spectra of fluctuating lift and cylinder displacement, respectively, characterise and distinguish the flow-induced vibrations. Figure 4(a–c) presents variations in the corresponding Strouhal numbers St $_{v}$ ( = f $_{\!v}$ D/U _∞) and St $_{w}$ ( = f $_{\!w}$ D/U _∞) with Re for various L/D values. The value of St $_{w}$ is smaller than St $_{v}$ for all the Re and L/D values examined. The value of St $_{w}$ with increasing Re generally decreases from 0.063 to 0.079 at Re = 1.97 × 10³ to 0.028–0.032 at Re = 1.18 × 10⁴ depending on L/D. The decrease with Re is rather mild in the GB compared with that in the DB. Although Re influences St $_{w}$ for all L/D values, the Re effect on St $_{w}$ is more palpable in the RF regime (L/D ≤ 3.0) than in the CF regime (L/D ≥ 3.5). Similarly, L/D predominantly affects St $_{w}$ for L/D ≤ 3.0, but not much for L/D ≥ 3.5. Why Re and L/D have greater effects on St $_{w}$ in the RF regime could be understood from the fact that the RF is characterised by the shear-layer reattachment where a small change in Re and/or L/D can greatly modify the shear-layer trajectory and hence the position of the shear-layer reattachment (Alam et al. Reference Alam, Moriya, Takai and Sakamoto2003; Xu & Zhou Reference Xu and Zhou2004; Alam Reference Alam2014), leading to changes in the flow-induced stiffness of the cylinder. On the other hand, the CF involves impingements of vortices shed from the upstream cylinder (Alam Reference Alam2016; Zafar & Alam Reference Zafar and Alam2018). The explanations of the mildly declining St $_{w}$ with Re are provided later.

Figure 4.

(a–c) Dependence of Strouhal numbers St $_{v}$ (dashed lines) and St $_{w}$ (solid slines) on Re and L/D. (d) Isocontours of St $_{w}$ corresponding to (a–c). The dashed line in (d) represents the boundary between GB and DB while the dashed-dotted line distinguishes RF and CF.

The value of St $_{v}$ displays a small peak at 2 × 10³ < Re < 6 × 10³ (depending on L/D) and a jump at 4 × 10³ < Re < 7 × 10³ (depending on L/D) for L/D = 2.0–4.5. The jump takes place at the onset of the violent vibration and is hence associated with the large-amplitude oscillation of the cylinder except for the smallest L/D = 2.0 where St $_{v}$ gradually waxes to 0.217 at Re = 5.36 × 10³ and then wanes to 0.149 at Re = 1.18 × 10⁴. For this L/D = 2.0, the flow may involve instabilities and strongly nonlinear interactions of the gap shear layers with the downstream cylinder. For two fixed cylinders, Xu & Zhou (Reference Xu and Zhou2004) observed that the behaviour of these shear layers varies with Re, either rolling off or reattaching onto the downstream cylinder. This results in more non-monotonic variations of St $_{v}$ at L/D = 2.0 than at other L/D values. In the GB (beyond the jump), St $_{v}$ wanes with Re for L/D ≤ 3.0 (RF) but more or less constant for L/D ≥ 3.5 (CF). Again, the St $_{v}$ is sensitive to both Re and L/D for L/D ≤ 3.0 but not for L/D ≥ 3.5, Re > 5 × 10³ where St $_{v}$ is comparable to the single cylinder value of 0.2 (Xu & Zhou Reference Xu and Zhou2004; Zhou et al. Reference Zhou, Alam, M. and Bai2009; Alam Reference Alam2014).

To better reflect the relationship between St $_{w}$ , Re and L/D, the isocontours of St $_{w}$ on a Re–L/D plane are displayed in figure 4(d). For a given L/D, St $_{w}$ decreases with Re in both DB and GB. The St $_{w}$ is more sensitive to Re and L/D in the RF regime than in the CF regime. In the GB, the contours are sparsely distributed, suggesting a weak dependence of St $_{w}$ on Re and L/D.

4.2.1. Reattachment flow

The power spectral density functions $E_{{C_{L}}}$ and E _Y of fluctuating lift forces and displacements are shown in figure 5(a,b) for Re = 1.97 × 10³–1.18 × 10⁴ at L/D = 3.0 corresponding to RF. To facilitate connections of St $_{v}$ and St $_{w}$ with the vibration responses, the corresponding A/D dependence on Re is illustrated in figure 5(c). For the DB (Re < 7 × 10³), the peak in St $_{v}$ is predominant over St $_{w}$ in both $E_{{C_{L}}}$ and E _Y . The St $_{w}$ peak in both $E_{{C_{L}}}$ and E _Y essentially heightens with increasing Re in the DB and becomes predominant over the St $_{v}$ peak in the GB (Re > 7 × 10³). This implies that the higher and lower ones are the vortex shedding and cylinder oscillation frequencies, respectively. In other words, the small-amplitude vibrations at Re < 7 × 10³ are not due to lock-in and hence cannot be said to be VIV but DB. In the GB (Re > 7 × 10³), the St $_{v}$ peak in $E_{{C_{L}}}$ and E _Y is hardly noticeable, small compared with St $_{w}$ peak because of the violent vibrations. One may argue that the feeble peak at St $_{v}$ may be because vortex shedding is suppressed or f $_{\!v}$ locks in with f $_{\!w}$ .

Figure 5.

Power spectral density functions (a) $E_{{C_{L}}}$ of lift force and (b) E _Y of displacement. (c) Relationship between A/D and Re. Here, L/D = 3.0 (RF).

4.2.2. Coshedding flow

As shown before, the relationships of St $_{v}$ and St $_{w}$ with Re or A/D bear some resemblances and disparities between the reattachment (L/D ≤ 3.0) and coshedding (L/D ≥ 3.5) flow regimes (figure 4). For example, one resemblance is that St $_{v}$ jumps at the onset of vibrations in both regimes. One disparity is that St $_{v}$ in the GB is insensitive to Re for the CF but declines for the RF. To explicate further, the dependencies of $E_{{C_{L}}}$ and E _Y on Re are presented in figure 6(a,b) for L/D = 4.0. Their connections with A/D can be viewed from figure 6(c).

Figure 6.

Power spectral density functions (a) $E_{{C_{L}}}$ of lift force and (b) E _Y of displacement. (c) Relationship between A/D and Re. Here, L/D = 4.0 (CF).

As shown in figure 6(a), the St $_{v}$ peak in the GB (Re > 5 × 10³) is rather pronounced in $E_{{C_{L}}}$ and does not change appreciably with increasing Re in view of peak height (energy) and frequency magnitude. The corresponding peaks for L/D = 3.0 were rather weak (figure 5 a). The difference arises from the distinct fluid dynamics involved in the RF and CF regimes. In the RF regime, the shear layers issuing from the upstream cylinder impinge or reattach on the downstream cylinder, and vortex shedding largely occurs behind the downstream cylinder. On the other hand, in the CF regime, in addition to the vortex shedding from the downstream cylinder, regular vortices shed from the upstream cylinder alternately impinge on the downstream cylinder (Zafar & Alam Reference Zafar and Alam2018; Zheng & Alam Reference Zheng and Alam2019). The fluctuating lift of the downstream cylinder is thus contributed by both impinging vortices and shed vortices from it. The St $_{v}$ peak is, therefore, rather stronger for the CF than for the RF (figures 5 a and 6 a). For the same reason, peaks in St $_{v}$ , albeit small, also appear in E _Y for L/D = 4.0 but not for L/D = 3.0 (figure 5 b, 6 b). Why St $_{v}$ is insensitive to Re in the CF ensues from the fact that the impinging vortices shed from the upstream cylinder trigger the vortex shedding from the downstream cylinder, where the upstream cylinder is subjected to the free-stream flow but the downstream cylinder is not.

Similar to L/D = 3.0 (figure 5), St $_{v}$ is greater than St $_{w}$ and undergoes a jump at the onset of vibrations (figure 6 a). In the DB, the St $_{v}$ peak is predominant over the St $_{w}$ peak in both $E_{{C_{L}}}$ and E _Y . At Re = 3.44 × 10³ where A/D has a small peak (figure 6 c), the St $_{v}$ peak is sharper and larger than any other peaks in the DB.

In the GB, the large-amplitude vibration of the cylinder is primarily driven by St $_{w}$ . In contrast, the oscillation amplitude in the DB is small but still considerable, influenced by both St $_{w}$ and St $_{v}$ in the lift force. This observation implies that both frequencies play an equally important role. This can be viewed from the low- and high-pass filtered signals of displacement and forces for DB (Re = 3.44 × 10³) and GB (Re = 7.36 × 10³) as presented in figures 7 and 8. The signals were filtered at a cutoff frequency of (f $_{\!w}$ + f $_{\!v}$ )/ 2. In the DB (figure 7), both low and high frequencies appear in the signals of displacement Y/D and lift force C _L , although the low-frequency component in C _L is less pronounced (figure 7 a). The low-pass (Y $_{w}$ /D) and high-pass (Y $_{v}$ /D) signals of Y/D display similar displacement amplitudes at both frequencies (figure 7 b). On the other hand, the low-pass (C $_{Lw}$ ) and high-pass (C $_{Lv}$ ) signals of C _L reveal that C $_{Lv}$ is dominant and exhibits more consistent peak amplitudes in the time domain than C $_{Lw}$ (figure 7 c). This further explains why the power spectrum peak is sharp in St $_{v}$ but blunt in St $_{w}$ (figure 6 a,b).

Figure 7.

Time histories of (a) raw Y/D and C _L , (b) low-pass (Y $_{w}$ /D) and high-pass (Y $_{v}$ /D) filtered Y/D and (c) low-pass (C $_{Lw}$ ) and high-pass (C $_{Lv}$ ) filtered C _L for L/D = 4.0, Re = 3.44 × 10³ (CF, DB).

Figure 8.

Time histories of (a) raw Y/D and C _L , (b) low-pass (Y $_{w}$ /D) and high-pass (Y $_{v}$ /D) filtered Y/D and (c) low-pass (C $_{Lw}$ ) and high-pass (C $_{Lv}$ ) filtered C _L for L/D = 4.0, Re = 7.36 × 10³ (CF, GB).

In the GB regime (figure 8), although the high-frequency component of the cylinder displacement is less apparent in the Y/D signal (figure 8 a), it is more evident in the filtered Y $_{v}$ /D signal, where the amplitude reaches approximately 0.07. However, this high-frequency component shows less consistent peak amplitudes over time (figure 8 b). In the case of lift forces, both C $_{Lw}$ and C $_{Lv}$ exhibit comparable amplitudes, but C $_{Lw}$ involves more consistent peak amplitudes in the time domain than C $_{Lv}$ (figure 8 c).

4.3. Flow structures: VIV or galloping?

Vortex-induced vibration and galloping are both types of flow-induced vibrations, but they arise from different mechanisms and exhibit distinct characteristics. The VIV results from a mutual lock-in between the vortex shedding and cylinder vibration frequencies. Galloping, in contrast, is an instability-driven vibration, where vortex shedding and cylinder vibration frequencies are not synchronised. To press home the point of whether f $_{\!v}$ locks in with f $_{\!w}$ , the PIV system was used to investigate vortex shedding from the vibrating cylinder.

4.3.1. Reattachment flow

Figure 9 presents the contours of normalised instantaneous vorticity $\omega _{{z}}^{*}(=\omega _{z}D/U_{\infty })$ for a half-cycle (t/T = 0 – 1/2) of the cylinder oscillation at Re = 7.36 × 10³. Seven instants (a–g) in t/T = 0 – 1/2 are evenly selected from the response curve shown in figure 9(h). The time histories of C _L (raw and low-pass filtered) are also included in figure 9(h) as C _L is more associated with alternate vortex shedding. To separate the C _L component from the vortex shedding, the raw C _L signal was low-pass filtered with a cutoff frequency of (f $_{\!w}$ + f $_{\!v}$ )/2. The low-pass filtered C _L has the same frequency as the Y/D signal, and the two signals are nearly antiphased. On the other hand, the raw signal displays high-frequency periodical changes associated with the vortex shedding. Here, f $_{\!w}$ and f $_{\!v}$ are 0.76 and 3.78 Hz, respectively, which gives f $_{\!v}$ /f $_{\!w}$ = 4.97.

Figure 9.

(a–g) Iso-contours of instantaneous vorticity $\omega _{z}^{*}$ in one half of the vibration cycle. The contour cutoff levels are ± 0.6 and the increment is 0.3. (h) Time histories of Y/D and C _L in one half of the vibration period, where the vertical dashed lines mark the time instants corresponding to plates (a–g). Here, L/D = 3.0 and Re = 7.36 × 10³ (RF).

The flow over the upstream cylinder generates shear layers in the gap between the cylinders (figure 9 a–g). These gap shear layers vigorously interact with the downstream cylinder, featuring alternate reattachment, impingement and switch from one side to the other. At t/T = 0 corresponding to Y/D = 0 (figure 9 a), vortex B2 stems from the reattached shear layer on the lower side, following a shed vortex A1 from the upper side. At the same time, a new vortex A2 is born on the upper side. As the cylinder moves upward at instant t/T = 1/12 (figure 9 b), A2 grows and appears closer to the cylinder while B2 moves downstream and is pinched off. A new vortex B3 is born on the lower side, making C _L negative. With the cylinder further moving upward (t/T = 1/6, figure 9 c), the upper gap shear reattaching on the front surface swerves down. The swerve splits the lower gap shear layer. The vortex B3 lying close to the cylinder rapidly grows in size, receiving vorticity from the split shear layer and the downstream-cylinder surface. On the other hand, A2 moving downstream is followed by an elongated shear layer from the upper side of the downstream cylinder. The growth of the elongated shear layer and the shift in the reattachment point (higher pressure) from the upper side to the front surface both cause C _L to increase (figure 9 h) (Alam et al. Reference Alam, Moriya, Takai and Sakamoto2003). When the cylinder reaches its maximum amplitude (t/T = 1/4, figure 9 d), all three shear layers – two gap shear layers and the downstream-cylinder-generated lower shear layer – flow over the lower side of the downstream cylinder. The flow over the lower side of the downstream cylinder is thus enhanced, which causes the declination of C _L . The vortex B3 is now fully developed, and the upper shear layer is further elongated. As the cylinder moves back (t/T = 1/3, figure 9 e), the two gap shear layers and the lower shear layer come close to each other, with the two gap shear layers enveloping the growing vortex B4. The envelopment leads to the formation of a large vortex on the lower side, hence a highly negative C _L (figure 9 h). The upper shear layer now rolls and forms a vortex A3. With the cylinder further moving down (t/T = 5/12 – 1/2, figure 9 f,g), vortex A3 is shed, followed by A4 on the upper side. The B4 and B5 vortices on the lower side follow suit. The two gap shear layers become symmetrical about the downstream cylinder. The growth of vortex B5 on the lower side keeps the C _L negative. In the entire half-cycle, a total of approximately five vortices – two (A2, A3) from the upper side and three (B2, B3, B4) from the lower side – shed from the downstream cylinder, which is echoed from the five local peaks and valleys in the C _L history. The vortex shedding frequency is thus different from the vibration frequency, which precludes the lock-in.

Figure 10.

(a–g) Instantaneous vorticity $\omega _{z}^{*}$ contours in half-vibration cycle. The contour cutoff levels are ± 0.6 and the increment is 0.3. (h) Time histories of Y/D and C _L in half-vibration period where the vertical dashed lines mark the time instants corresponding to plates (a–g). Here, L/D = 4.0 and Re = 7.4 × 10³ (CF).

4.3.2. Coshedding flow

For the CF at L/D = 4.0, Re = 7.36 × 10³, the involved flow– or vortex–cylinder interactions are presented in figure 10(a–g) for a half-oscillation cycle. It is worth noting that f $_{\!w}$ and f $_{\!v}$ are 0.62 and 3.54 Hz, respectively, for this case, which yields f $_{\!v}$ /f $_{\!w}$ = 5.7. Figure 10(h) displays the Y/D and C _L (raw and low-pass filtered) histories, where the vertical dashed lines represent the instants of snapshots in figure 10(a–g). Again, Y/D and low-pass filtered C _L are nearly antiphased. As explained above, the flow pattern is coshedding, each cylinder shedding vortices. When the downstream cylinder is inline with the upstream one (t/T = 0, figure 10 a), four vortices (D1, D2, C1, C0) feature the cylinder gap, and a binary vortex D0 + B1 sheds from the lower side of the downstream cylinder. As the cylinder moves upward (t/T = 1/12, figure 10 b), the gap vortex D1 meets with the newly growing vortex B2 on the lower side of the downstream cylinder. Zafar & Alam (Reference Zafar and Alam2018) and Zheng & Alam (Reference Zheng and Alam2019) showed that vortex B2 springs from the impulse of vortex D1. In other words, the latter triggers the growth of the former. This combined vortex (B2 and D1), appearing on the lower side of the downstream cylinder, makes C _L highly negative. With the cylinder moving further upward (t/T = 1/6, figure 10 c), D1 and B2 shed as a binary vortex D1 + B2 from the lower side while vortex A2 following the impulse of C1 grows from the upper side. The latter causes an increase in C _L from (b) to (c). When the cylinder is at its extreme position (t/T = 1/4, figure 10 d), the C1 and A2 vortices combine and shed as a binary vortex C1 + A2 while vortex D2 pairs with vortex B3 growing from the lower side. They finally shed as a binary vortex D2 + B3 at instant t/T = 1/3 (figure 10 e). Similarly, as time elapses, binary vortices shed from both the upper and lower sides (figure 10 e–g). This suggests that, similar to two tandem fixed cylinders, the vortices from the upstream cylinder trigger the vortex shedding from the downstream cylinder, which leads to binary vortices prevailing in the wake. The violent vibration is not due to the lock-in of vortex shedding with the cylinder oscillation.

Figure 11.

(a–g) Instantaneous vorticity $\omega _{z}^{*}$ contours in half-vibration cycle. The contour cutoff levels are ±0.6 and the increment is 0.3. (h) Time histories of Y/D and C _L in half-vibration period where the vertical dashed lines mark the time instants corresponding to plates (a–g). Here, L/D = 4.0 and Re = 1.0 × 10⁴ (CF).

When Re is increased to 1.0 × 10⁴ with the same L/D = 4.0 (figure 11), f $_{\!w}$ is 0.76 Hz and f $_{\!v}$ is 4.91 Hz, resulting in a f $_{\!v}$ /f $_{\!w}$ of 6.4. As before, Y/D and the low-pass filtered C _L are antiphased, while the raw C _L signal (figure 11 h) and instantaneous vortex structures (figure 11 a–g) indicate the shedding of multiple vortices within one half-cycle of cylinder oscillation. At t/T = 0 (figure 11 a), three vortices (D1, C1, C0) are present in the cylinder gap. A binary vortex D0 + B1 sheds from the lower side of the downstream cylinder, and vortex A1 is about to shed from the upper side. As the cylinder moves from t/T = 0 to 1/12 (figure 11 b), the gap vortex C0 couples with A1, shedding as a combined vortex C0 + A1 from the upper side. Simultaneously, the lower gap vortex D1 combines with a new vortex B2 from the lower side of the downstream cylinder, similar to the vortex interaction observed at Re = 7.36 × 10³ (figure 10 b). The positions of vortices D1 and B1, near the wake centreline, facilitate their amalgamation. At the peak displacement of the cylinder (figure 11 d), the increased lateral separation between the two cylinders creates larger lateral spaces for the coupled vortices D2 and B3. Therefore, these vortices do not merge but retain their individual identities as the cylinder moves downward (figure 11 d–f). At the same time, vortex merging takes place from the upper side. These observations again substantiate that vortices from the upstream cylinder trigger the vortex shedding from the downstream cylinder, leading to the prevalence of binary vortices in the wake. Conversely, when the lateral spacing between the cylinders is large during vibration, some gap vortices fail to merge with vortices shed from the downstream cylinder, maintaining their individual identities.

4.4. Lift force and phase lag

Lift force and its phase lag with the cylinder displacement are two governing parameters that determine the cylinder vibration. The lift force represents the degree of the excitation force involved in the vibration. On the other hand, the phase lag determines the two force components associated with flow-induced energy dissipation (damping) and flow-induced potential energy (stiffness), respectively. Phase lags of less than 90° and greater than 90° render negative and positive flow-induced stiffnesses to the system, respectively. Figure 12(a–c) shows the fluctuating lift coefficient $C^{\prime}_{L}$ as a function of Re and L/D. The value of $C^{\prime}_{L}$ exhibits a complex relationship with Re and L/D. For each L/D, there is a small peak at Re = 2 × 10³–4 × 10³. This peak is linked to the small peak in A/D (figure 3). The value of $C^{\prime}_{L}$ also jumps at the boundary (Re = 4 × 10³–7 × 10³) between the DB and GB for L/D = 2.0–4.5. In the GB, $C^{\prime}_{L}$ diminishes with increasing Re although A/D increases or levels off (figure 3). It raises the following questions: What is the insight into increasing A/D when $C^{\prime}_{L}$ is decreased? What is the parameter causing A/D to increase despite the decreasing $C^{\prime}_{L}$ ? The same questions also apply to the nearly invariant $C^{\prime}_{L}$ in the GB (Re = 4 × 10³–12 × 10³) for L/D = 5.0 – 6.0 where A/D grows with increasing Re. The questions are answered in the next sections. At L/D = 3.5, the jump in $C^{\prime}_{L}$ at the vibration inception is rather wild. Note that, although L/D = 3.5 is in the RF regime for the fixed cylinders, a small-amplitude vibration or any excitation in the flow may transmute the RF into the CF for the same L/D (Sakamoto & Haniu Reference Sakamoto and Haniu1988; Alam Reference Alam2021).

Figure 12.

(a–c) Dependence of fluctuating lift force coefficient $C^{\prime}_{L}$ on Re and L/D. (d) Iso-contours of $C^{\prime}_{L}$ corresponding to (a–c). The dashed curve in (d) represents the boundary between GB and DB while the dashed-dotted line distinguishes RF and CF.

The value of $C^{\prime}_{L}$ in the DB is largely contributed by alternate vortex shedding for both reattachment and coshedding regimes (figure 5 a, 6 a, 9 h, 10 h, 11 h). On the other hand, that in the GB is largely bestowed by the cylinder vibration for the reattachment regime (figure 5 a) and by both cylinder vibration and vortex shedding for the coshedding regime (figure 6 a). The dependence of $C^{\prime}_{L}$ on Re and L/D can further be viewed from the isocontours of $C^{\prime}_{L}$ (figure 12 d). In the (CF, DB) regime, a local $C^{\prime}_{L}$ peak emerges at Re = 2.96 × 10³ – 4.08 × 10³ for L/D = 3.5–5.0, corresponding to the local A/D peak in figure 3(b,c). In the GB, the maximum $C^{\prime}_{L}$ occurs at L/D = 3.5, Re = (6–8) × 10³, which corresponds to the CF near the boundary between the RF and CF regimes.

The value of $C^{\prime}_{L}$ has two components: one associated with the vortex shedding (high frequency, f $_{\!v}$ ) and the other with the cylinder oscillation (low frequency, f $_{\!w}$ ). The value of C _L is, therefore, decomposed into the low- and high-frequency components (i.e. C $_{Lw}$ and C $_{Lv}$ ) using low- and high-pass filters, respectively. The corresponding fluctuating lift forces are estimated as $C^{\prime}_{Lw}$ and $C^{\prime}_{Lv}$ , and their dependencies on Re and L/D are presented in figure 13. Interestingly, in the DB for all L/D values (both RF and CF), the $C^{\prime}_{Lw}$ value is not significantly sensitive to Re and is much smaller than $C^{\prime}_{Lv}$ . The latter is more sensitive to Re. In the GB, $C^{\prime}_{Lw}$ – significantly greater than $C^{\prime}_{Lv}$ – decreases with Re for the RF (L/D = 2.0–3.0). The cases for L/D = 3.5–4.5, CF with smaller L/D, are different, where $C^{\prime}_{Lw}$ in the GB increases with Re for L/D = 4.0 and 4.5 but remains nearly invariant for L/D = 3.5. The $C^{\prime}_{Lv}$ is now larger for L/D = 3.5–4.5 than for L/D = 2.0–3.0, as the former is associated with the CF where the impingement of gap vortices on and vortex shedding from the cylinder itself both contribute to the fluctuating lift force. For L/D = 3.5–4.5, $C^{\prime}_{Lv}$ diminishes with increasing Re as A/D increases. When the vortex shedding frequency differs from the cylinder vibration frequency, the increasing vibration amplitude adversely affects the vortex shedding, which is weakened and less organised. For the same reason, $C^{\prime}_{Lv}$ in the GB declines for L/D = 2.5–3.0 but mildly grows for L/D = 2.0. The $C^{\prime}_{Lv}$ growth for the latter case is due to the increase in Re as A/D is nearly invariant. In the cases of L/D = 5.0–6.0, CF with larger L/D, $C^{\prime}_{Lw}$ in the GB increases with Re because of increasing A/D while $C^{\prime}_{Lv}$ declines because of the degradation of vortex shedding at higher A/D, overwhelming the contribution of the increasing Re. Overall, for the CF with the GB (L/D = 3.5–6.0), the $C^{\prime}_{Lv}$ declines with increasing Re and L/D because of weaker impingement of gap vortices and the degradation of vortex shedding from the downstream cylinder.

Figure 13.

Dependence on Re and L/D of the fluctuating lift force coefficients (a–c) $C^{\prime}_{Lw}$ and (d–f) $C^{\prime}_{Lv}$ associated with cylinder oscillation (low frequency) and vortex shedding (high frequency), respectively.

The phase lag $\phi$ is estimated from cross-correlations $R_{{F_{L}},Y}$ between the F _L and Y signals captured simultaneously (Qin, Alam & Zhou Reference Qin, Alam and Zhou2017), viz.,

(4.1) \begin{equation} R_{{F_{L}},Y} (\tau)= \frac{\overline{\{Y(t)\}\{F_{L}(t+\tau)\}}} {[\{\overline{Y(t)\}^{2}}]^{1/2} [\overline{\{F_{L}(t+\tau)\}^{2}}]^{1/2}},\end{equation}

representing the cross-correlation coefficient as a function of time delay $\tau$ . It reaches its maximum when the two signals are in phase, which can be achieved by varying $\tau$ . The value of $\tau$ corresponding to the maximum $R_{{F_{L}},Y}$ is scaled with a phase lag $\phi$ in angular degrees in one vibration cycle. The phase lag between the lift and displacement indicates vortex shedding timing relative to cylinder motion (Chen et al. Reference Chen, Ji, Alam, Xu, An, Tong and Zhao2022). In the case of structural stiffness existing, the following conditions satisfy the VIV regime (Bhatt & Alam Reference Bhatt and Alam2018; Lin & Alam Reference Lin and Alam2024):

(4.2) \begin{equation} \phi \lt 90^{\circ} \quad{\textrm {with}}\; f_v = f_o \lesssim f_n \quad ({\textrm {IB}}),\end{equation}

(4.3) \begin{equation} \phi \gt 90^{\circ} \quad\;\; {\textrm {with}}\; f_v = f_o \gtrsim f_n \quad ({\textrm {LB}}),\end{equation}

and

(4.4) \begin{equation} \phi \gt 90^{\circ} \quad{\textrm {with}}\; f_v = f_o \gg f_n \quad {\textrm {and/or}} \quad f_v \gt f_o \gtrsim f_n \quad ({\textrm {DB}}).\end{equation}

When the structural damping is zero, a jump in $\phi$ from 0° to 180° takes place at the resonance (f $_{\!v}$ = f $_{\!w}$ = f _n ) condition. When f _n = 0 (no structural stiffness), the condition in (4.2) cannot be met as the vortex shedding frequency or oscillation frequency cannot be negative. It is, nevertheless, possible to meet the conditions in (4.3) and (4.4). In other words, the IB is not possible, but the LB and DB may be. Indeed, although the structural stiffness is zero, the resultant or effective stiffness including the wake-induced stiffness is non-zero, positive as is shown later. In such a case, all branches are hypothetically possible.

In the DB, the predominant vortex shedding frequency is generally higher than the predominant oscillation frequency. The lift and displacement signals each displays both f $_{\!v}$ and f $_{\!w}$ , with f $_{\!v}$ predominant in the lift signal and both f $_{\!w}$ and f $_{\!v}$ predominant in the displacement signal, as can be seen in the DB in figures 5 and 7. The condition f $_{\!v}$ > f $_{\!w}$ ≳ f _n ( = 0) is met in the small-amplitude vibration regime, which justifies the name ‘desynchronisation branch’.

As the cylinder in the present experiment has zero structural stiffness, it brings in curiosity about how the phase lag contributes to vibrations in the GB and DB. Since lift and displacement signals each comprises both f $_{\!v}$ and f $_{\!w}$ , the signals were low- and high-pass filtered with a cutoff frequency of (f $_{\!w}$ + f $_{\!v}$ )/2. We then estimated phase lags $\phi_{w}$ and $\phi_{v}$ associated with the low- and high-pass filtered signals, respectively. The parameter $\phi_{w}$ and $\phi_{v}$ can be considered phase lags between the lift and displacement associated with cylinder vibration and vortex shedding, respectively. Figure 14 shows the relationships of $\phi_{w}$ and $\phi_{v}$ with Re and L/D. The values of $\phi_{w}$ and $\phi_{v}$ are always greater than 90°, regardless of Re and L/D (i.e. regardless of DB, GB, RF and CF). The value of $\phi_{v}$ mostly appears close to 180° in both DB and GB for all L/D values (figure 14 d–f). On the other hand, $\phi_{w}$ is smaller than 180° in the DB where $C^{\prime}_{Lw}$ is essentially small, but lies close to 180° in the GB where $C^{\prime}_{Lw}$ is large (figures 13 a–c, 14 a–c). The departure of $\phi_{w}$ from 180° in the DB is larger at smaller Re, increasing from L/D = 2.0–3.0 to L/D = 3.5–4.5 and then to L/D = 5.0–6.0. It is plausible to expect the phase lag to be 180° as no structural damping is implemented, given that there are no springs in the model. Although no structural damping is set, the air bearings – used to hold and slide the model – provide negligibly small Coulomb damping. This damping dissipates energy from the vibrating system, resulting in a phase lag between the lift force and displacement. The phase lag depends on how much energy is dissipated compared with the system energy. The smaller the vibration amplitude and/or frequency, the smaller the system energy. At a smaller vibration amplitude and frequency (DB), where $C^{\prime}_{Lw}$ is small, the energy dissipation, albeit small, can become comparable to the system energy at a low amplitude and frequency. This leads to a phase lag of less than 180° in the DB. The phase lag declines from 180° with decreasing Re as the system energy increases with decreasing Re because of the higher oscillation frequency. Further insight into the phase lag dependence on Re and vibration amplitude is provided later.

Figure 14.

Dependence on Re and L/D of (a–c) phase lags $\phi_{w}$ and (d–f) $\phi_{v}$ associated with the cylinder vibration and vortex shedding frequencies, respectively.

4.5. Drag force

In the RF regime, the mean drag coefficient $\overline{C}_{D}$ of the downstream cylinder of two tandem fixed cylinders is negative, depending on Re and L/D, indicating that this cylinder experiences a thrust force (Zdravkovich & Pridden 1997; Alam et al. Reference Alam, Moriya, Takai and Sakamoto2003; Sumner Reference Sumner2010). The relationships of $\overline{C}_{D}$ with Re and L/D are shown in figure 15(a–c). At L/D = 2.0 – 3.0, $\overline{C}_{D}$ is largely negative in the DB and positive in the GB, with a jump at the border between the DB and GB. It is more negative for L/D = 2.0 at Re > 4 × 10³ than for L/D = 2.5 and 3.0. For L/D = 3.5 – 4.5, $\overline{C}_{D}$ peaks at Re = 2.9 × 10³–4.1 × 10³ corresponding to the small peak in A/D. Again, jumps in $\overline{C}_{D}$ take place at the inception of the large-amplitude vibration. In the GB, $\overline{C}_{D}$ is less sensitive to Re and L/D. Similar observations are made for L/D = 5.0–6.0, where $\overline{C}_{D}$ is further less sensitive to Re and L/D. The value of L/D = 5.0 has smaller values of $\overline{C}_{D}$ for Re < 3.4 × 10³ than L/D = 5.5 and 6.0 as A/D at the former spacing was smaller than the latter. In the GB, $\overline{C}_{D}$ is smaller in the reattachment regime than in the coshedding regime. Overall, a higher A/D is accompanied by a higher $\overline{C}_{D}$ . The correlation of $\overline{C}_{D}$ with Re and L/D is presented by the isocontours of $\overline{C}_{D}$ in figure 15(d). The $\overline{C}_{D}$ appears negative for small A/D in the RF regime. In the DB, $\overline{C}_{D}$ peaks at the same location as $C^{\prime}_{L}$ does. The $\overline{C}_{D}$ is minimum at L/D = 2.0, Re = 4.7 × 10³. The $\overline{C}_{D}$ is less sensitive to L/D and Re in the GB with CF.

Figure 15.

(a–c) Dependence of the time-averaged drag force coefficient $\overline{C}_{D}$ on Re and L/D. (d) Iso-contours of $\overline{C}_{D}$ corresponding to (a–c). The dashed curve in (d) represents the boundary between GB and DB while the dashed-dotted line distinguishes RF and CF.

As proved above, the fluctuating lift force at the oscillation frequency grows with increasing A/D while the increase in A/D reduces the fluctuating lift force at the vortex shedding frequency. This is because the cylinder oscillation enhances the fluctuating flow velocity at the cylinder oscillation but weakens the vortex shedding from the oscillating cylinder. It is worth understanding how A/D is linked to the fluctuating drag force. Figure 16(a–c) shows the variations of fluctuating drag force coefficient $C^{\prime}_{D}$ with Re and L/D. At L/D = 2.0–3.0 (figure 16 a), $C^{\prime}_{D}$ is small in the DB, waning with Re before jumping at the onset of vibration. The scenario is different at L/D = 3.5–4.5 where the jump in $C^{\prime}_{D}$ associated with that in A/D is rather weak for L/D = 3.5 and 4.0 but vanishes for L/D≥ 4.5 (figure 16 b,c). Particularly, L/D and Re have almost no effects on $C^{\prime}_{D}$ for L/D = 5.0–6.0 although Re is found to have a significant effect on A/D. In the GB, $C^{\prime}_{D}$ is not as large as A/D, compared with those in the GB. The $C^{\prime}_{L} , C'_{Lv}$ and $\overline{C}_{D}$ all displayed a peak corresponding to the small peak in A/D in the DB; the cause of the small peaks is still not well understood. On the other hand, $C^{\prime}_{D}$ does not show this behaviour. This implies that $C^{\prime}_{D}$ is quite insensitive to A/D in the coshedding regime. The isocontours of $C^{\prime}_{D}$ are illustrated on Re–L/D plane in figure 16(d). In the DB, the value of $C^{\prime}_{D}$ is weakly dependent on Re and L/D. In the GB, the value of $C^{\prime}_{D}$ decreases with increasing L/D for a given Re. At L/D = 5.0–6.0, $C^{\prime}_{D}$ is insensitive to Re and L/D. The $C^{\prime}_{D}$ is more sensitive to A/D in RF and less sensitive to A/D in CF.

Figure 16.

(a–c) Dependence of the fluctuating drag force coefficient $C^{\prime}_{D}$ on Re and L/D. (d) Iso-contours of $C^{\prime}_{D}$ corresponding to (a–c). The dashed curve in (d) represents the boundary between GB and DB while the dashed-dotted line distinguishes RF and CF.

4.6. Flow-induced effective stiffness k_e

The cylinder exhibits flow-induced vibrations in the absence of springs (zero structural stiffness). It is a well-accepted argument that vibrations are not possible without stiffness (neutral equilibrium). The function of stiffness is nothing but generating a restoring force that is in antiphase with the structure displacement from its equilibrium position. Since there is no physical stiffness in the system, the only way of having the stiffness is the flow-induced effective stiffness resulting from the wake- and motion-induced lift force generated on the cylinder to sustain the cylinder motion. When $\phi$ is the phase lag between the lift force and displacement, the lift force can be expressed as

(4.5) \begin{equation} F_{L0}\sin (\omega t+\phi ) = F_{L0}\, {\cos}\phi \sin \omega t + F_{L0}\, {\sin} \phi \cos \omega t, \end{equation}

where the first and second terms on the right-hand side are force components in phase with the cylinder displacement and velocity, respectively. Equation (3.1) can, therefore, be written as

(4.6) \begin{equation} m\ddot{Y}+\left(c-\frac{F_{L0}\sin \phi }{\omega Y_{0}}\right)\dot{Y}+\left(k-\frac{F_{L0}\cos \phi }{Y_{0}}\right)Y=0. \end{equation}

Equation (4.6) reflects that $-{F_{L0}\sin \phi }/{\omega Y_{0}}$ is the flow-induced effective damping c _e , and $-{F_{L0}\cos \phi }/{Y_{0}}$ is the flow-induced effective stiffness k _e , both resulting from fluid–structure interactions. We can, therefore, write

(4.7) \begin{equation} c_{e}=-\frac{F_{L0}\sin \phi }{\omega Y_{0}}\quad \textrm{and}\quad k_{e}=-\frac{F_{L0}\cos \phi }{Y_{0}}. \end{equation}

Interestingly, k _e is determined by the ratio of $F_{L0}$ and Y ₀, while $\phi$ must be greater than 90° for k _e to be positive. Therefore, $\phi$ > 90° is always observed for this investigation of the cylinder with no structural stiffness (figure 14), leading to a positive flow-induced stiffness for vibration generation. For a steady-state vibration, the energy transferred from the fluid to the structure equals the energy dissipated by structural damping. That is, the net damping of the cylinder system is zero, which yields

(4.8) \begin{equation} \begin{array}{c} c-\frac{F_{L0}\sin \phi }{\omega Y_{0}}=0\\[10pt] \Rightarrow \sin \phi =\frac{c\omega Y_{0}}{F_{L0}}\\[10pt] \begin{array}{l} \Rightarrow \phi =\sin ^{-1}\left(c\omega Y_{0}/F_{L0}\right)\\[10pt] =\pi -\sin ^{-1}\left(c\omega Y_{0}/F_{L0}\right). \end{array} \end{array} \end{equation}

The equation indicates that $\phi$ is a function of c, $\omega$ , Y ₀ and F _L0. Given that c is constant for a system, $\phi$ is determined by $\omega$ Y ₀/F _L0 that is the ratio of the cylinder velocity amplitude to the lift force amplitude. A larger $\omega$ Y ₀ compared with F _L0 can make $\phi$ close to 90° while a smaller $\omega$ Y ₀ can yield $\phi$ close to 0° or 180°. By the same token, a very small c again makes $\phi$ close to 0° or 180° unless $\omega$ Y ₀/F _L0 is very high.

In our experiments, structural viscous damping is zero but there is a negligibly small Coulomb damping. The difference between the viscous damping and the Coulomb damping is that viscous damping force ( $c\dot{Y}$ ) is directly proportional to the cylinder velocity and dissipates energy as

(4.9) \begin{equation} W_{d}=\pi c\omega Y_{0}^{2}, \end{equation}

while the Coulomb damping force (F _c ) is constant and dissipates energy as

(4.10) \begin{equation} W_{c}=4F_{c}Y_{0}. \end{equation}

Equating Equations (4.9) and (4.10), we can obtain an equivalent viscous damping constant for the Coulomb friction as

(4.11) \begin{equation} c_{cv}=4F_{c}/\left(\pi \omega Y_{0}\right). \end{equation}

Equation (4.9–4.11) offer a couple of pieces of crucial information into the insights into the characteristics of viscous damping and Coulomb damping and their roles in the vibrations. Firstly, viscous damping constant c remains constant, regardless of Y ₀ and $\omega$ , while the equivalent Coulomb damping constant c $_{cv}$ is inversely proportional to Y ₀ and $\omega$ . The higher the values of Y ₀ and $\omega$ , the smaller the value of c $_{cv}$ , given that F _c does not change with Y ₀ and $\omega$ . Secondly, dissipated energy is directly proportional to $Y_{0}^{2}$ and Y ₀ for viscous damping and Coulomb damping, respectively. This implies that energy dissipation is higher for viscous damping, compared with Coulomb damping. Plugging c $_{cv}$ in (4.11) into (4.8), i.e. c = c $_{cv}$ , we can find the phase lag for a system undergoing Coulomb damping as

(4.12) \begin{equation} \phi =\pi -\sin ^{-1}\left\{\left(4/\pi \right)\left(F_{c}/F_{L0}\right)\right\}. \end{equation}

Interestingly, the phase lag is now dependent on F _L0 that is a representation of $C^{\prime}_{Lw}$ . This explains why $\phi_{w}$ is nearly 180° in the GB where the vibration amplitude as well as $C^{\prime}_{Lw}$ is high. At the same time, it also explains why $\phi_{w}$ is less than 180° in the DB (small-amplitude vibration) where $C^{\prime}_{Lw}$ is small (figure 14 a–c). As $C^{\prime}_{Lv}$ is large in both DB and GB, $\phi_{v}$ stays around 180°, regardless of vibration branches (figure 13 d–f, 14 d–f).

Equation (4.12) can also be used to calculate $\phi_{w}$ and $\phi_{v}$ directly from $C^{\prime}_{Lw}$ and $C^{\prime}_{Lv}$ , respectively, without measuring displacement. The F _c value, albeit very small, can be measured from the experimental set-up or from a measurement of phase lag and lift force. The values of $\phi$ $_{w}$ and $\phi_{v}$ were obtained from (4.1) (figure 14), and the lift force amplitude F _L0 is known from figure 13. Thus, in (4.12) F _c is the only unknown on the right-hand side and can be readily determined by inverting the equation, i.e. $F_{c}=(\pi F_{L0}\sin \phi )/4$ . Obtaining the F _c value for L/D = 5.5, Re = 2 × 10³, we, using Eq. (15), numerically estimated $\phi_{w}$ and $\phi_{v}$ for L/D = 2.5, 4.5 and 6.0 and compare them with those estimated from the lift and displacement signals in figure 14. We also measured the Coulomb forces at other (L/D, Re) = (4.0, 2 × 10³), (5.0, 2 × 10³) and (5.5, 3.4 × 10³) values, which had deviation of less than 3.6 % compared with the case at L/D = 5.5 and Re = 2 × 10³. The resulting error is small, lying in the experimental uncertainties.

Figure 17 compares $\phi_{w}$ and $\phi_{v}$ between estimated and measured values. The estimated $\phi_{v}$ values concur well with the measured ones (figure 17 d–f) while small deviations between the estimated and measured $\phi_{w}$ values are discernible, especially for L/D = 6.0 in the DB. The deviations are ascribed to the dim peak at the oscillation frequency in E _Y and hence to small F _L0 at the oscillation frequency. Conversely, the vortex shedding frequency has an obvious peak in $E_{{C_{L}}}$ for all Re values examined, the deviations in $\phi_{v}$ are thus negligible.

Figure 17.

Comparisons of the phase lags (a–c) $\phi_{w}$ and (d–f) $\phi_{v}$ between calculations from (4.12) and from the measured lift and displacement signals.

Now, the effective stiffness can be rewritten as an integral term associated with Y

(4.13) \begin{equation} k_{e}=-\frac{2}{nTY_{0}^{2}}\int _{t}^{t+nT}F_{L}Y{\textrm d}t', \end{equation}

where T is the oscillation period, n is the number of periods considered for the integration, t is the initial time of integration and $t'$ is the integration variable. The effective stiffness is derived from the force component that is in antiphase with the displacement. While Y ₀ and T are based on linear assumptions, the term F _L Y – since both F _L and Y are measured directly – naturally captures any nonlinearities in the wake force–displacement relationship.

The value of k _e , which is a function of Re and L/D, represents the flow-induced effective restoring force, which is proportional to the displacement. The above equations are admissible for sinusoidal motion at a single frequency. As proved above, the C _L and Y/D signals each contains two frequencies: the cylinder oscillation frequency and the vortex shedding frequency. The corresponding stiffnesses k $_{ew}$ and k $_{ev}$ were calculated from the low- and high-pass filtered signals of C _L and Y/D.

Figure 18(a–c) presents the dependence of k $_{ew}$ on Re and L/D. The value of k $_{ew}$ generally grows linearly with Re, except for L/D = 2.0–3.5 where the shear-layer reattachment, detachment and switch are involved. At L/D = 2.0, the value of k $_{ew}$ linearly increases to a peak of 40.7 at Re = 6.7 × 10³ before linearly declining. For Re > 9 × 10³, A/D is more or less constant and $C^{\prime}_{Lw}$ rapidly declines (figures 3 a and 13 a). Given $C^{\prime}_{Lw}=({2F_{L0w}}/{\rho U_{\infty }^{2}Dl})\propto ({F_{L0w}})/({Re^{2}})$ , the lift force amplitude F $_{\!L0w}$ associated with f $_{\!w}$ may decrease if $C^{\prime}_{Lw}$ decreases more rapidly than the increase of Re ². A smaller F $_{\!L0w}$ yields a smaller k $_{ew}$ magnitude (4.7), vindicating the decrease of k $_{ew}$ with Re. On the other hand, for L/D = 2.5 and 3.0, in the GB, although A/D generally increases, the gentle $C^{\prime}_{Lw}$ declination could not overwhelm the increase of Re ². The value of k $_{ew}$ thus increases with Re, albeit with a much smaller slope than that in the DB. The behaviour of k $_{ew}$ at L/D ≥ 3.5 (coshedding regime) differs from that at L/D = 2.0–3.0. For L/D ≥ 3.5, k $_{ew}$ generally increases with Re because of a large increase in F $_{\!L0w}$ , overwhelming the increase of A/D. A break in the linear change in k $_{ew}$ is identified at Re = 5.7 × 10³ for L/D = 3.5, which is a footprint of the transmutation of the RF to the CF as discussed before. A global view of k $_{ew}$ dependencies on L/D and Re is presented in figure 18(d) in terms of isocontours of k $_{ew}$ . The parameter k $_{ew}$ is more sensitive to L/D and Re in the reattachment regime (L/D < 3.5) than in the coshedding regime (L/D ≥ 3.5). In the latter regime, k $_{ew}$ is largely influenced by Re, not appreciably by L/D.

Figure 18.

(a–c) Dependence on Re and L/D of added stiffness k $_{ew}$ associated with the vibration frequency. (d) Iso-contours of k $_{ew}$ corresponding to (a–c). The dashed curve in (d) represents the boundary between GB and DB while the dashed-dotted line distinguishes RF and CF.

The relationship between k $_{ew}$ and Re thus differs between the L/D values in view of the k $_{ew}$ –Re slopes. Figure 19 summarises and compares k $_{ew}$ –Re relationships between the L/D values examined. There are two slopes or two linear relationships for each L/D: one in the DB and the other in the GB. Radical differences in the slopes between the DB and GB are observed for L/D = 2.0–3.5.

Figure 19.

Relationships between k $_{ew}$ and Re for various L/D values. The solid and dashed lines represent k $_{ew}$ in the DB and GB, respectively.

The empirical correlation between Re and k $_{ew}$ could be obtained from curve fits to the data presented in figure 19

(4.14a) \begin{equation} \left.\begin{array}{l} k_{ew} = 0.00805Re-13.66 \quad \text{for}\; Re = 1900-6680\\[8pt] k_{ew}=-0.00278Re+59.25 \quad\text{for}\; Re = 6680-12000 \end{array}\right\}L/D=2.0, \end{equation}

(4.14b) \begin{equation} \left.\begin{array}{l} k_{ew} = 0.00357Re-5.17\quad \text{for}\; Re = 1900-7000\\[8pt] k_{ew} = 0.00166Re+16.44\quad\text{for}\; Re = 7000-12000 \end{array}\right\}L/D=2.5, \end{equation}

(4.14c) \begin{equation} \left.\begin{array}{l} k_{ew} = 0.00301Re-1.91\quad \text{for}\; Re = 1900-7000\\[8pt] k_{ew} = 0.00323Re+0.086\quad\text{for}\; Re = 7000-12000 \end{array}\right\}L/D=3.0, \end{equation}

(4.14d) \begin{equation} \left.\begin{array}{c} k_{ew} = 0.00244Re-1.83 \quad\text{for}\; Re = 1900-5700\\[8pt] k_{ew} = 0.00311Re-10.27\quad\text{for}\; Re = 5700-12000 \end{array}\right\}L/D=3.5, \end{equation}

(4.14e) \begin{equation} \left.\begin{array}{c} k_{ew} = 0.00162Re-1.42\quad\text{for}\; Re = 1900-5000\\[8pt] k_{ew} = 0.00310Re-8.05 \quad\textrm{for}\; Re = 5000-12000 \end{array}\right\}L/D=4.0, \end{equation}

(4.14f) \begin{equation} \left.\begin{array}{c} k_{ew} = 0.00165Re-1.32 \quad\text{for} \; Re = 1900-4400\\[8pt] k_{ew} = 0.00299Re-7.03 \quad\text{ for}\; Re = 4400-12000 \end{array}\right\}L/D=4.5, \end{equation}

(4.14g) \begin{equation} \left.\begin{array}{c} k_{ew} = 0.00136Re-0.53\quad\text{for}\; Re = 1900-3750\\[8pt] k_{ew} = 0.00294Re-6.76\quad\text{for}\; Re = 3750-12000 \end{array}\right\}L/D=5.0, \end{equation}

(4.14h) \begin{equation} \left.\begin{array}{c} k_{ew} = 0.00118Re+0.26\quad\text{for}\; Re = 1900-3750\\[8pt] k_{ew} = 0.00292Re-7.06\quad\text{for}\; Re = 3750-12000 \end{array}\right\}L/D=5.5, \end{equation}

(4.14i) \begin{equation} \left.\begin{array}{c} k_{ew} = 0.00123Re+0.001\quad\text{for}\; Re = 1900-3750\\[8pt] k_{ew} = 0.00284Re-7.26\quad\text{for}\; Re = 3750-12000 \end{array}\right\}L/D=6.0. \end{equation}

The value of k $_{ew}$ is linearly related to Re for all examined L/D = 2.0 – 6.0, each L/D displaying two slopes associated with the DB (solid line) and GB (dashed line). All k $_{ew}$ lines have positive slopes except for L/D = 2.0 in the GB. To highlight the relationships of the slope ${\partial k_{ew}}/{\partial\! Re}$ with L/D, the two ${\partial k_{ew}}/{\partial\! Re}$ values for each L/D are presented in figure 20 as a function of L/D. The dramatic and mild differences in slope occur at L/D = 2.0 and 3.0, respectively. For the CF (L/D > 3.0), the slope is almost invariant with L/D in the GB, while it decreases with L/D (>3) in the DB. Therefore, a general equation for k $_{ew}$ is proposed, viz.

Figure 20.

Dependence of slope $({\partial \textit{k}_{\textit{ew}}}/{\partial \textit{Re}})$ on L/D.

(4.15) \begin{equation} \left.\begin{array}{l} k_{ew} = 0.003Re+C_{1}\,\,\,\,\left(\textrm{GB}\right)\\[8pt] k_{ew} = \left(-0.00058L/D+0.0044\right)Re+C_{2}\,\,\,\,\left(\textrm{DB}\right) \end{array}\right\}\,\left(\text{coshedding}\,\text{flow}\right)\!, \end{equation}

where C ₁ and C ₂ are constants which could be obtained from the boundary conditions.

Equation (4.15) specifies the relationship between k $_{ew}$ and Re for the CF in the DB and GB. On the other hand, ${\partial k_{ew}}/{\partial\! Re}$ is highly sensitive to L/D for both DB and GB. Particularly, ${\partial k_{ew}}/{\partial\! Re}$ for the GB changes from 0.00166 (positive) at L/D = 2.5 to −0.00278 (negative) at L/D = 2.0 while that for the DB increases rapidly when L/D is decreased from 3.0 to 2.0.

4.7. Role of flow-induced effective stiffness in determining vibration frequency

As observed in figure 4, St $_{w}$ declines with increasing Re and the declination is sensitive to both L/D and GB (i.e. Re). To further show the relationship of St $_{w}$ with L/D and Re, St $_{w}$ could be shown as a function of Re while the natural frequency f $_{\!nw}$ associated with k $_{ew}$ and corresponding reduced velocity U $_{rw}$ can be expressed as

(4.16) \begin{equation} St_{w}=\frac{f_{w}D}{U_{\infty }}=\frac{f_{w}\rho D^{2}/\mu }{Re}, \end{equation}

(4.17) \begin{equation} f_{nw}={1}/{2\pi }\sqrt{k_{ew}/m}, \end{equation}

and

(4.18) \begin{equation} \begin{array}{l} U_{rw}= \displaystyle\frac{U_{\infty }}{f_{nw}D}= \displaystyle\frac{2\pi U_{\infty }}{D\sqrt{k_{ew}/m}}=\displaystyle\frac{2\pi \rho U_{\infty }D/\mu }{\rho D^{2}\sqrt{k_{ew}/m}/\mu }\\[16pt] \def\lumina\hspace*{1.8pc} =\displaystyle\frac{2\pi \mu Re}{\rho D^{2}\sqrt{k_{ew}/m}} .\end{array} \end{equation}

Note that St $_{w}$ is estimated from the power spectrum of the directly measured displacement signal while U $_{rw}$ and f $_{\!nw}$ are estimated from the fluid stiffness k $_{ew}$ . Multiplying (4.16) with (4.18), we can obtain

(4.19) \begin{equation} St_{w}\times U_{rw}=\frac{2\pi f_{w}}{\sqrt{k_{ew}/m}}=\frac{f_{w}}{f_{nw}}. \end{equation}

The value of $St_{w}\times U_{rw}$ essentially represents the ratio of directly measured oscillation frequency and estimated natural frequency associated with the flow-induced effective stiffness. Figure 21(a–c) presents the variations of $St_{w}\times U_{rw}$ with L/D and Re. It is observed that $St_{w}\times U_{rw}$ is greater than 1.0 and generally decreases with Re in the DB while approaching 1.0 in the GB for all L/D values examined. The results indicate that the oscillation frequency in the GB is the same as the natural frequency associated with the effective stiffness caused by the flow. In the DB, $St_{w}\times U_{rw}$ > 1.0 implies f $_{\!w}$ > f $_{\!nw}$ where the quiescent-fluid added mass and Coulomb damping c $_{cv}$ play a role in modifying f $_{\!nw}$ as k $_{ew}$ is small. For a circular cylinder, the quiescent-fluid added mass equals the displaced fluid mass m _d by the cylinder. Taking m _d and c $_{cv}$ into account in the calculation of f $_{\!nw}$ , we can obtain

Figure 21.

Dependence on Re and L/D of $St_{w}$ × $U_{rw}$ : (a–c) when only $k_{ew}$ is considered in $U_{rw}$ , and (d–f) when $k_{ew}$ , m _d and $c_{cv}$ are all considered $U_{rw}$ .

(4.20) \begin{equation} f_{nw}=\frac{1}{2\pi }\sqrt{\frac{k_{ew}}{m+m_{d}}+\frac{c_{cv}\dot{Y}_{0}}{\left(m+m_{d}\right)Y_{o}}}=\frac{1}{2\pi }\sqrt{\frac{k_{ew}}{m+m_{d}}+\frac{4F_{c}}{\pi \left(m+m_{d}\right)Y_{0}}}. \end{equation}

Now f $_{\!nw}$ involves m _d , F _c and Y ₀. Equation (4.20) reflects that m _d and F _c may have a greater role in f $_{\!nw}$ when k $_{ew}$ and Y ₀ are small (i.e. in the DB) while their role is negligible when k $_{ew}$ and/or Y ₀ are large (i.e. in the GB). Considering f $_{\!nw}$ in Eq. (23), $St_{w}\times U_{rw}$ is recalculated and presented in figure 21(d–f). Now all $St_{w}\times U_{rw}$ data collapse into 1.0 for all Re and L/D values, regardless of small- or large-amplitude vibrations. This further substantiates that m _d and F _c can modify f $_{\!nw}$ for a small k $_{ew}$ and Y ₀ but not for a large k $_{ew}$ and Y ₀. The collapse of $St_{w}\times U_{rw}$ data onto 1.0 suggests that f $_{\!w}$ = f $_{\!nw}$ ; that is, the cylinder oscillation occurs at the aerodynamic natural frequency corresponding to the aerodynamic stiffness. This outcome can now further explain why St $_{w}$ declines with increasing Re in figure 4.

Combining (4.16) and (4.17)

(4.21) \begin{equation} St_{w}=\frac{f_{w}D}{U_{\infty }}=\frac{\rho D^{2}\sqrt{k_{ew}/m}}{2\pi \mu Re}, \end{equation}

where k $_{ew}$ and Re are the variables on the right-hand side. It has been proved that k $_{ew}$ has a linear relationship with Re. Equation (4.21) echoes that St $_{w}$ is directly and inversely proportional to $\sqrt{k_{ew}}$ (not k $_{ew}$ ) and Re, respectively. The value of St $_{w}$ , therefore, declines with Re.

Similarly, k $_{ev}$ associated with vortex shedding frequency is presented in figure 22(a–c), estimated from the lift force and displacement signals associated with vortex shedding frequency using (4.13). Again, k $_{ev}$ escalates nonlinearly with Re where the escalation is larger in the coshedding regime (figure 22 b,c) than in the reattachment regime (figure 22 a). The k $_{ev}$ far exceeds k $_{ew}$ , the former being one order of magnitude larger than the latter in the GB (figures 18 and 22). This also explains why the vibration amplitude or lift force associated with the vortex shedding frequency is small in the GB. The corresponding $St_{v}\times U_{rv}$ values taking into account m _d and F _c are depicted in figure 22(d–f). The $St_{v}\times U_{rv}$ values nestle around 1.0 although they are slightly higher at smaller Re. This further confirms that the cylinder oscillation at the shedding frequency also follows the corresponding flow-induced added stiffness as does the low-frequency oscillation.

Figure 22.

Dependence on Re and L/D of (a–c) added stiffness k $_{ev}$ associated with the vortex shedding frequency, and (d–f) St $_{v}$ × U $_{rv}$ , where k $_{ev}$ , m _d and c $_{cv}$ are considered in U $_{rv}$ .