3.3.1 Bistable flow regime

The bistable flow phenomenon associated with the intermittent occurrence of reattachment and co-shedding flows has been reported for two rigid cylinders (e.g. Igarashi Reference Igarashi1982). We discuss here the bistable flow when the downstream cylinder is flexibly supported.

Figure 6 presents $E_{u1}$ , $E_{u2}$ and $E_{Y}$ of simultaneously captured $u_{1}$ , $u_{2}$ and $Y$ signals ( $L/d=3.0$ , $d/D=1.0$ and $U_{r}=11.0$ ). The plot of $E_{u1}$ (figure 6 a) displays one pronounced peak at 22.0 Hz or $St=0.171$ , while that of $E_{u2}$ (figure 6 b) shows two at 17.6 Hz $(St=0.137)$ and 22.0 Hz $(St=0.171)$ . Obviously, the single $St$ in the gap corresponds to the co-shedding mode. Under the reattachment flow mode, there should be no discernible peak in the spectra measured between the cylinders (e.g. Alam et al. Reference Alam, Moriya, Takai and Sakamoto2003; Xu & Zhou Reference Xu and Zhou2004). The occurrence of two predominant frequencies is ascribed to vortex shedding from the downstream cylinder under the flow modes of reattachment and co-shedding, respectively, or the bistable flow. The plot of $E_{Y}$ (figure 6 c) shows one pronounced peak at $f_{n}~(=11.7~\text{Hz})$ and also two minor peaks at the shedding frequencies of 17.6 Hz and 22.0 Hz.

Figure 6. (a–c) Power spectral density functions $E_{u1}$ , $E_{u2}$ and $E_{Y}$ of simultaneously captured $u_{1}$ , $u_{2}$ and $Y$ , respectively.

3.3.2 Violent vibration regime

In view of the objectives of this work, we present first data showing the dependence of the vibration response on $d/D$ and $L/d$ and thus classify the flow regimes with and without vibration. To understand the initiation and mechanism of the vibration, we illustrate one representative case, i.e.  $L/d=2.0$ and $d/D=0.4$ , which occurs near the centre of the vibration regime in the $L/d$ – $d/D$ plane. It is worth pointing out that we have actually examined a number of other cases, which are not shown because of the same flow physics.

3.3.2.1 Connection between vibration and vortex shedding

Representative $E_{u2}$ and $E_{Y}~(d/D=0.4,~L/d=2.0)$ are shown in figure 7(a,b). The $f/f_{n}=f_{v}/f_{n}$ corresponding to the peaks in the spectra (figure 7 a) depends on $U_{r}$ , as shown in figure 7(c). The corresponding vibration response is given in figure 7(d). Figure 7 allows us to identify the sources of structural vibration, as indicated on the left side of figure 7(a). A small peak that occurs over $U_{r}=4.5{-}5.3$ in figure 7(d) corresponds to $f_{v}/f_{n}=1.0$ in both $E_{u2}$ and $E_{Y}$ (figure 7 a,b), apparently due to the lock-in of vortex shedding with the first-mode structural vibration or the resonance. At $U_{r}=3.8$ , one peak only occurs at $f/f_{n}=0.875$ in figure 7(a) and two at $f/f_{n}=0.875$ and 1.0 in figure 7(b), implying that $f_{v}$ is not locked in with $f_{n}$ . Naturally, the cylinder vibration is insignificant or negligibly small at both $f_{v}$ and $f_{n}$ . At $U_{r}=5.3$ , again two peaks are observed at $f/f_{n}=1.0$ and 1.19 in figure 7(a,b). The peak at $f/f_{n}=1.0$ is less pronounced in $E_{u2}$ and more in $E_{Y}$ than its counterpart at $f/f_{n}=1.19$ , suggesting an intermittent lock-in in the transition between the lock-in (VE) and non-lock-in. The peak at $f/f_{n}=1.0$ indicates the occurrence of lock-in and the other at $f/f_{n}=1.19$ corresponds to natural vortex shedding. The intermittent lock-in at the boundary between the lock-in and non-lock-in regimes is a well-established phenomenon (Leontini et al. Reference Leontini, Stewart, Thompson and Hourigan2006). For a single circular cylinder, Pan, Cui & Miao (Reference Pan, Cui and Miao2007) found that $C_{L}$ and $Y/D$ appeared less periodic at the end of the lock-in range due to the intermittent occurrence of lock-in.

Figure 7. Power spectral density functions (a) $E_{u2}$ and (b) $E_{Y}$ , and dependence on $U_{r}$ of (c)  $f_{v}/f_{n}$ , and (d) $A/D$ , for $L/d=2.0$ and $d/D=0.4$ .

As $U_{r}$ goes beyond the lock-in range (figure 7 a,c), $f_{v}/f_{n}\neq 1.0$ , rising monotonically following $St=0.2$ until $U_{r}=8.5$ . There is little vibration over $5.3<U_{r}<8.5$ . Similarly to the non-lock-in at $U_{r}=3.8$ , one and two peaks are seen in $E_{u2}$ and $E_{Y}$ , respectively. The $A/D$ value becomes quite significant at $8.5\leqslant U_{r}<9.8$ (figure 7 d); the second peak corresponding to $f_{v}$ is more appreciable (figure 7 b), though $E_{u2}$ does not show any peak at $f_{v}/f_{n}=1.0$ (figure 7 a). The results indicate that the vibration is initiated by vortex shedding from the cylinder, not due to excitation at $f_{n}$ . More evidence will be provided later. On the other hand, when $A/D$ is large enough at $U_{r}>9.8$ , $E_{u2}$ display two prominent peaks (figure 7 a), corresponding to $f_{v}/f_{n}=1.0$ and $St=0.2$ (figure 7 c). The latter is maintained up to $U_{r}=14.9$ and then $f_{v}$ is locked in with the third harmonic of $f_{n}$ for $U_{r}>14.9$ , where $A/D$ rises slowly.

3.3.2.2 Downstream cylinder vibration effect on flow

The flows at $U_{r}=4.9$ and $U_{r}=14.9$ are examined to understand the differences between the flow structures associated with the VE and galloping vibrations. For each $U_{r}$ , using the $Y$ signal as a reference signal, we calculated the phase-averaged vorticity contours, based on 64 images, at eight instants for each period $T$ of the cylinder oscillation. The contours of the eight instants are presented for $U_{r}=4.9$ in figure 8(a–h). The correspondence between time instant and cylinder displacement is given in figure 8(i). Here $t/T=0$ refers to the instant when the cylinder crosses the equilibrium position $Y/D=0$ , moving upwards. The vortex formation process may be divided into three stages: growth, saturation and decay (Jeon & Gharib Reference Jeon and Gharib2004). The three stages are also seen presently when the downstream cylinder is vibrating, as discussed below.

Figure 8. (a–h) Phase-average vorticity $\unicode[STIX]{x1D714}_{z}^{\ast }~(=\unicode[STIX]{x1D714}_{z}D/U_{\infty })$ contours in one vibration cycle. Solid and dotted contour lines stand for clockwise and anticlockwise vorticities. The contour cut-off levels are $\pm 0.2$ and the increment is 0.2. (i) Variation in $Y/D$ with $t/T$ where the time instants are specified by the dashed lines corresponding to (a–h). Here $L/d=2.0$ , $d/D=0.4$ and $U_{r}=4.9$ .

VE vibration. Under the VE vibration $(U_{r}=4.9)$ , the lower shear layer in the gap at $t/T=0$ reattaches onto the lower front side of the downstream cylinder and splits into two (figure 8 a). One moves downstream, and the other flows over the front surface and meets the upper shear layer streaming on the upper side of the downstream cylinder. Four vortices, i.e. B1, B2, A2 and A3, are separated from the downstream cylinder. Between the instants in figure 8(a) and figure 8(b), B2 and A3 are close to the flow separation point and grow due to the continued addition of vorticity from the boundary layers. At the instant in figure 8(b), B1 is about to separate to become a free vortex (Gerrard Reference Gerrard1966). As $Y/D$ reaches its maximum (figure 8 c, $t/T=1/4$ ), the upper shear layer in the gap reattaches, and part of the reattached shear layer flows along the lower side of the cylinder (figure 8 c). B1 is now completely separated with a tail. Between $t/T=1/12$ and $1/4$ , B2 moves downstream appreciably, but A3 does not, indicating vortex shedding from the lower side. Consequently, a new vortex B3 emerges on the lower side (figure 8 c,d). Being closer to the separation point than B2, B3 accumulates vorticity and grows during $t/T=1/4{-}5/12$ . Since vortex shedding from the lower side takes place during the half period (figure 8 a–e), A3 is not advected downstream appreciably.

As the cylinder crosses $Y/D=0$ and moves downwards, vortex shedding from the upper side starts to occur. A3 moves downstream and grows during $t/T=1/2{-}3/4$ . A new vortex A4 emerges at $t/T=11/12$ . The lower shear layer reattaches again at $t/T=3/4$ , and A2 is detached with a tail from the shear layer. It may be inferred that vortex shedding occurs largely from the lower side for $Y/D>0$ and from the upper side for $Y/D<0$ . Therefore, in each oscillation cycle, two free vortices B1 and A2 are completely separated from the shear layers (figure 8 b,f) and two new vortices B3 and A4 emerge. As such, the dynamics of the wake behind the vibrating cylinder is similar to what is found in a classical von Kármán street, exhibiting the classical 2S vortex formation mode (Williamson & Roshko Reference Williamson and Roshko1988). Although the reattached shear layers sweep over the front surface of the downstream cylinder, the flow in the gap between the cylinders is mostly stagnant and the lateral positions of the gap shear layers change little during the oscillation of the downstream cylinder. The observation suggests that the shear layer separated from the upstream cylinder may not produce a significant effect on the lift and hence vibration of the downstream cylinder. Rather, the vibration is generated due to vortex shedding from the downstream cylinder.

Figure 9. (a–h) Phase-average vorticity $\unicode[STIX]{x1D714}_{z}^{\ast }~(=\unicode[STIX]{x1D714}_{z}D/U_{\infty })$ contours in one cycle. Solid and dotted contour lines stand for clockwise and anticlockwise vorticities. The contour cut-off levels are $\pm 0.2$ and the increment is 0.2. (i) Variation in $Y/D$ with $t/T$ where the time instants are specified by the dashed lines corresponding to (a–h). Here $L/d=2.0$ , $d/D=0.4$ , $A/D=0.48$ and $U_{r}=14.9$ .

Galloping vibration. The flow structure exhibits a difference at $U_{r}=14.9$ ( $L/d=2.0$ and $d/D=0.4$ ) when galloping occurs. Figure 9 presents the phase-averaged vorticity contours during one cycle of oscillation. Recall that the third harmonics lock-in occurs at $U_{r}=14.9{-}17.7$ for $L/d=2.0$ and $d/D=0.4$ . The corresponding flow structure will be explored here. At $t=0~(Y/D=0)$ , the lower shear layer separated from the upstream cylinder reattaches on the downstream cylinder while rolling up in the gap, a small part of the reattached flow joining the lower shear layer of the downstream cylinder (figure 9 a). As the cylinder moves upwards, the upper gap shear layer reattaches on the cylinder (figure 9 b) and swerves downwards without reattachment when $Y/D$ is large enough (figure 9 c). Both upper and lower gap shear layers join the lower shear layer around the downstream cylinder (figure 9 b,c). Unlike A2 and B2, a new vortex A3 emerging between the instants in figure 9(b) and figure 9(c) originates from the shear layer over the upper side of the cylinder. This vortex does not seem to interact directly with the shear layers separated from the upstream cylinder. A2 occurs on the wake centreline $(y^{\prime }/D=0)$ at the instant in figure 9(c). B2 below $y^{\prime }/D=0$ follows the cylinder movement and moves towards $y^{\prime }/D=0$ from the instant in figure 9(a) to figure 9(c). While leading B2 marginally at the instant in figure 9(a), A2 is $2.25D$ downstream of B2 in figure 9(c). Both A2 and B2 occur near $y^{\prime }/D=0$ in figure 9(d), and their streamwise separation remains unchanged from the instant in figure 9(c) to figure 9(d).

A3 stays away from the wake centreline during the cylinder downward motion (figure 9 c–g). New vortices B3 and A4 emerge at the instant in figure 9(d), and B3 leads A4. They interact vigorously with each other (figure 9 d–f). A4 on the upper side follows the cylinder downward motion, approaching $y^{\prime }/D=0$ . In figure 9(h), both A4 and B3 are on $y^{\prime }/D=0$ . It is interesting to note that A3 and B4 are formed from the shear layers, pushed away from the upstream cylinder wake, at the instants in figure 9(b,c) and (f,g), respectively, and both stay away (approximately $y/D=\pm 1$ ) from the wake centreline, not directly interacting with the gap shear layers. On the other hand, B2 and A2 originate at least partially from the gap shear layers and end up on the wake centreline. So do B3 and A4. Thus, a total of six vortices are formed during one cycle of the cylinder oscillation, accounting for the third harmonic lock-in, $f_{v}/f_{n}=3.0$ .

While vortices B2, A2, B3 and A4 near the centreline decay rapidly in terms of the maximum vorticity concentration, A3 and B4, far away from the centreline, do so rather slowly. The wake is thus dominated by the latter type of vortex produced alternately following the up-and-down cylinder oscillation. It will be shown later that the cylinder receives energy from flow (positive work) during the shedding process of the former type of vortex. On the other hand, the flow absorbs energy from the cylinder (negative work) during the shedding of the other type of vortex. This difference in energy transfer may also explain why the former type of vortex decays rapidly and the latter survives longer.

The vortex shed from the downstream cylinder swerves downwards at the instants in figure 9(a,b) when the cylinder moves upwards, vortex shedding appearing symmetric in figure 9(c) when the cylinder velocity is zero (maximum $Y/D$ ). As the cylinder moves downwards (figure 9 c,d), the vortex shed from the downstream cylinder is deflected upwards, with a maximum deflection occurring at the instant in figure 9(e) when the cylinder velocity is maximum. The two gap shear layers pass over the lower and upper sides of the downstream cylinder that reaches the maximum (figure 9 c) and minimum $Y/D$ (figure 9 g), respectively. Obviously, the alternating reattachment and switch of the two gap shear layers from one side to the other follow the cylinder oscillation, the switch frequency being synchronized with the cylinder oscillation frequency. Accordingly, there is a radical change in the pressure distribution around the downstream cylinder, which will be discussed in § 3.4. A smaller $d/D$ facilitates the gap shear layer switch, as the shear layers can attach at a small incident angle. On the other hand, a large $d/D$ leads to the occurrence of the reattachment at a relatively large incidence angle. Thus, a small $d/D$ is associated with a violent vibration (figure 5).

The gap flow switch has been suggested as an excitation mechanism of vibration for a flexible cylinder placed behind another of identical diameter (Zdravkovich Reference Zdravkovich1988; Knisely & Kawagoe Reference Knisely and Kawagoe1990; Ruscheweyh & Dielen Reference Ruscheweyh and Dielen1992; Dielen & Ruscheweyh Reference Dielen and Ruscheweyh1995). The downstream cylinder oscillation could not be excited if placed in line with the upstream cylinder but could be excited if given an initial displacement of greater than 0.2 $D$ (Zdravkovich Reference Zdravkovich1974) or a flow incidence angle of more than $10^{\circ }$ . The oscillation is generated due to the occurrence of two flow modes, one with a strong gap flow producing a large inward lift force and the other without gap flow (shear–shear layer reattachment) associated with a negligible lift force (Zdravkovich Reference Zdravkovich1988; Dielen & Ruscheweyh Reference Dielen and Ruscheweyh1995). For a stationary downstream cylinder, the former mode may take place when the lateral distance between the two cylinders and the incidence angle exceed critical values. For an elastic downstream cylinder, a strong gap flow persists longer when the downstream cylinder moves towards the wake centreline than moving away from the centreline, and the mode without gap flow starts later as the cylinder shifts away from the centreline. This means a spatial phase lag of flow switching with respect to the lateral motion of the downstream cylinder (Ruscheweyh & Dielen Reference Ruscheweyh and Dielen1992; Dielen & Ruscheweyh Reference Dielen and Ruscheweyh1995). As a result, a phase lag is generated between $C_{L}$ and $Y$ , which makes the vibration sustainable, corresponding to a positive energy transfer from the flow into the cylinder. Dielen & Ruscheweyh (Reference Dielen and Ruscheweyh1995) proposed the above scenario based on the data of two stationary cylinders, without any information from simultaneously measured $C_{L}$ and $Y/D$ . Their scenario is to a certain extent consistent with the present observation, explaining why the downstream vibration is excited for $d/D<1.0$ but not for $d/D=1.0$ . While their scenario suggests only the gap flow switching, our data show unequivocally that the interaction of the two gap shear layers with the downstream cylinder plays a key role for the generation of the structural vibration.

Based on the quasi-steady theory of galloping, galloping cannot be generated for an axisymmetric body, e.g. a circular cylinder (Bearman et al. Reference Bearman, Gartshore, Maull and Parkinson1987). As such, one question arises: Why does galloping takes place for the circular cylinder placed in the wake of another? The two cylinders in tandem are connected by shear layers in the reattachment regime. Then, the effective shape of the two-cylinder system is no longer axisymmetric; hence the two cylinders may be prone to generating galloping vibrations. Given that the upstream cylinder is not fixed, galloping may also occur for the upstream cylinder in the reattachment regime (Bokaian & Geoola Reference Bokaian and Geoola1984b ). In the co-shedding regime, the upstream cylinder does not experience galloping but the downstream cylinder does (Bokaian & Geoola Reference Bokaian and Geoola1984a ,Reference Bokaian and Geoola b ). This is because the upstream cylinder may be considered to be isolated and is now axisymmetric. On the other hand, the downstream cylinder is not axisymmetric with respect to local approaching flow since the flow between the cylinders is non-uniform. Therefore, the galloping motion of the downstream circular cylinder, observed in both reattachment and co-shedding regimes, is not a violation of the galloping theory.

Wake evolution. The wake pattern is sketched over a longer range of $x/D$ in figure 10 so that the vortex generation and dynamics can be better illuminated. The cylinder position corresponds to the instant in figure 9(h) and the vortices are shed at the instants in figure 9(a–h). The lateral separations from the centreline $(y/D=0)$ of vortices A3 and B1 are $y/D=\pm 1$ , as discussed before. The vortices are placed in the wake based on their positions. The streamwise separations between free vortices A4 and B3, B3 and A3, and A3 and B2 are 2.4 $D$ , 1.8 $D$ , and 1.0 $D$ , respectively, which are extracted from figure 9(h,g,f), respectively. Also, the streamwise distances between B2 and A2, A2 and B1, and B1 and A1 are 2.4 $D$ , 1.8 $D$ and 1.0 $D$ , respectively, obtained from figure 9(d,c,b) in view of the observation that the convection velocity of vortices changes little beyond $x/D=5.7$ (Lin & Hsieh Reference Lin and Hsieh2003). It seems that vortices A1, B1 and A2 form one set, whilst B2, A3 and B3 form another. Interestingly, the latter set originates from the cylinder surface during the cylinder moving from the centreline to the top (figure 9 a–c) and the associated shedding occurs during the cylinder motion from the top to the centreline. Similarly, the former set of vortices emerges in the next half cycle of the cylinder motion. The vortices on the centreline decay rapidly, as discussed earlier. Being close to each other, B2 and A3 may be considered to be one pair. Although the pattern appears like a $2\text{P}+2\text{S}$ pattern, the downstream wake survives in a 2S pattern.

Figure 10. Sketch of the vortex street behind the cylinders, following figure 9. P and S stand for a vortex pair and a single vortex, respectively. The red dashed line encircles the vortices shed in one complete oscillation cycle.

Figure 11 presents the instantaneous vorticity contours for one oscillation cycle of the cylinder at $U_{r}=12.6$ . Note that $f_{v}/f_{n}$ at $U_{r}=12.6$ is 2.6 (figure 7 c), which is a non-integer number, that is, the downstream cylinder oscillation and vortex shedding do not have a fixed phase shift. Thus phase averaging is inappropriate. Between the instants in figure 11(a) and figure 11(b), concentration A2 tends to break away from the upper shear layer, while B2 grows in size. As the cylinder reaches its maximum amplitude (figure 11 c), B2 breaks away and A3 grows, accompanied by A4. From figure 11(a) to figure 11(c), two vortices (A2 and B2) are formed, with vorticity originated largely from the upper shear layer, which is connected to the upward movement of the cylinder. Similarly, from figure 11(c) to figure 11(g), three vortices A3, B3 and A4 are generated, and vorticity largely originates from the lower shear layer. B4 is about to break away at the instant in figure 11(h) and $t/T=1.0$ (not shown). Thus, a total of 5+ vortices are separated from the cylinder, joining the vortex street, in one period of oscillation. The result conforms to $f_{v}/f_{n}\approx 2.6$ in a 2S mode. No pairing is observed among these vortices. It is worth pointing out that another frequency at $f_{v}/f_{n}=1.0$ is observed at $U_{r}=12.6$ (figure 7 a), which is due to the cylinder oscillation as the very near wake is directed upwards with the cylinder moving downwards and vice versa.

Figure 11. (a–h) Instantaneous vorticity $\unicode[STIX]{x1D714}_{z}^{\ast }~(=\unicode[STIX]{x1D714}_{z}D/U_{\infty })$ contours in one vibration cycle. Solid and dotted contour lines stand for clockwise and anticlockwise vorticities. The contour cut-off levels are $\pm 0.6$ and the increment is 0.3. (i) Variation in $Y/D$ with $t/T$ where the time instants are specified by the dashed lines corresponding to (a–h). Here $L/d=2.0$ , $d/D=0.4$ and $U_{r}=12.6$ .

Sen & Mittal (Reference Sen and Mittal2015) numerically studied the in-line and transverse free vibrations of a square cylinder. For $m^{\ast }=10$ and 20, the classical galloping vibration occurred and a total of six vortices were generated within an oscillation cycle of low oscillation amplitude, forming a 3(2S) pattern. The pattern changed to 2P+2S when the transverse displacement surpassed a threshold. It may be inferred that, with an increase in $U_{r}$ from 12.6 to 14.9, the transverse displacement of the cylinder increases and the transition from the 2S to 2P+2S pattern might take place.

In order to understand further the flow evolutions under the VE, no vibration and violent vibration, HT2 was traversed from $x/D=4$ to 30 in the wake for three different $U_{r}=4.7$ (VE), 8.3 (no vibration) and 14.9 (violent vibration). The power spectra of the HT2 signals are presented in figure 12. When the cylinder does not vibrate (figure 12 b), the only peak at $f_{v}$ in $E_{u2}$ decays rapidly as $x/D$ increases. On the other hand, in the VE regime, the peak is relatively wider and decays less rapidly (figure 12 a). For the violent vibration case (figure 12 c), the peak at $f_{v}/f_{n}=3$ wanes rapidly with increasing $x/D$ . However, the peak at $f/f_{n}=1.0$ grows downstream. The observation is internally consistent with the finding from figure 9 that the vortices connected to shear layer impingement decay rapidly and those shed when the cylinder is far away from the centreline persist longer.

Figure 12. Power spectral density function $E_{u2}$ for (a) VE at $U_{r}=4.7$ , (b) no vibration at $U_{r}=8.3$ , and (c) galloping at $U_{r}=14.9$ . Here $L/d=2.0$ and $d/D=0.4$ .

Figure 13. (a–e) Phase-averaged vorticity contours and pressure coefficient $C_{P\text{-}pa}$ around the downstream cylinder in one half cycle of the cylinder oscillation. The contour cut-off levels are $\pm 0.2$ and the increment is 0.2. (f) Variation with time in the displacement $Y/D$ and lift force coefficient $C_{L}$ over the half cycle. Here $L/d=2.0$ , $d/D=0.4$ and $A/D=0.48$ .

3.4.1 Work done

Whether a vibration is sustainable depends on the energy transfer between fluid and cylinder, which is subsequently linked to the way a vortex is shed from the cylinder. It is important to understand the work done by $C_{L}$ during cylinder vibration and vortex shedding, separately. To this end, $C_{L}$ is low- and high-pass-filtered at a cut-off frequency of 15 Hz. Figure 15 presents the temporal evolution of instantaneous and cumulative work, $W_{fi}~(=C_{L}\times d(Y/D))$ and $\sum W_{fi}~(=\sum C_{L}\times d(Y/D))$ , done in two cycles of oscillations, where subscripts 1 and 2 denote the low- and high-pass-filtered $C_{L}$ , respectively, used in calculating $W_{fi}$ or $\sum W_{fi}$ . Thus $W_{fi}$ , $W_{fi,1}$ and $W_{fi,2}$ represent the total work done, work done from $C_{L}$ at the vibration frequency, and work done from $C_{L}$ at higher frequencies, respectively. Positive and negative $W_{fi}$ indicate flows acting to excite and to dampen the vibration, respectively.

Figure 15. (a) Time histories of $Y/D$ , filtered $C_{L}$ and instantaneous work done $W_{fi}$ in two cycles. (b,c) Variations in $W_{fi,2}$ , $\sum W_{fi}$ , $\sum W_{fi,1}$ and $\sum W_{fi,2}$ in (b) two cycles and (c) 15 cycles. (d) Sketch for the signs of work done in one half cycle of the cylinder oscillation. Here $L/d=2.0$ , $d/D=0.4$ and $A/D=0.48$ .

As shown in figure 15(a), $C_{L}$ decreases to zero as the cylinder moves upwards from $Y/D=0$ to 0.31, resulting in positive but decreasing $W_{fi,1}$ (process A). As the cylinder continues its upward movement till the top position where $d(Y/D)=0$ , $W_{fi,1}$ becomes negative (process B). Interestingly, the gap shear layers are involved in the positive $W_{fi,1}$ and the vortices formed when the cylinder is far away from $Y/D=0$ are connected to the negative $W_{fi,1}$ . Next, as the cylinder moves downwards from the top, $W_{fi,1}$ remains positive and reaches its maximum during this quarter of the period (process C). Evidently, there is one half cycle of oscillation in hysteresis due to a phase lag between $C_{L}$ and $Y/D$ , which results from the interaction between the shear layers, separated from the upstream cylinder, and the downstream cylinder as observed in the temporal evolution of the flow structure in figures 9 and 13. A summary sketch of the work done is presented in figure 15(d). While the cylinder moves upwards to $Y/D=0.31$ , the work done is positive and the cylinder absorbs energy from the flow (process A) when the shear layer reattachment takes place and switches from one mode to the other (figure 13). From $Y/D=0.31$ to its maximum, the work done is negative and the flow receives energy from the vibrating cylinder. However, when moving from the top to $Y/D=0$ , the cylinder gains energy from the flow for the entire process C, the maximum energy transfer rate occurring at $Y/D=0.19$ . Interestingly, the energy the cylinder receives is mostly (approximately 75 % of the total energy transfer) transferred at $Y/D<0.31$ where the gap shear layers interact vigorously with the downstream cylinder. It seems plausible that the shear layer reattachment and switch are the main sources to generate the cylinder vibration. The $\sum W_{fi,1}$ calculated from the low-pass-filtered $C_{L}$ is similar to $\sum W_{fi}$ calculated from the unfiltered $C_{L}$  (figure 15 b). The $\sum W_{fi,2}$ , calculated over two cycles, is very small, approximately 0.08, compared with $\sum W_{fi,1}~({\approx}2.0)$ . That is, energy transferred from the fluid to the cylinder is largely contributed from $\sum W_{fi,1}$ . The high-frequency modulation in $W_{fi,2}$ represents the shedding frequency. The work done over a longer time (figure 15 c) reveals that $\sum W_{fi,2}$ , though positive, is also very small, approximately $(1/30)\sum W_{fi,1}$ , suggesting a negligibly small contribution to the vibration from the lift associated with higher frequencies.

3.4.2 Added mass, added damping, and energy transfer model

The transverse free vibration of a cylinder in a cross-flow may be governed by the following equation (Bearman Reference Bearman1984; Govardhan & Williamson Reference Govardhan and Williamson2000; Williamson & Govardhan Reference Williamson and Govardhan2004)

(3.1) $$\begin{eqnarray}m\ddot{Y} +c{\dot{Y}}+kY=F_{L},\end{eqnarray}$$

where $m$ is the total oscillating cylinder mass, $c$ is the structural damping, $k$ is the spring stiffness and $F_{L}$ is the time-dependent fluid force in the transverse direction (lift force). The overdot represents a derivative with respect to time $t$ . The measured transverse force is often approximated by a sinusoidal function. Thus the motion and force may be given by

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle Y(t)=Y_{0}\sin \unicode[STIX]{x1D714}t, & \displaystyle\end{eqnarray}$$

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle F_{L}(t)=F_{L0}\sin (\unicode[STIX]{x1D714}t+\unicode[STIX]{x1D719}), & \displaystyle\end{eqnarray}$$

where $Y_{0}$ is the amplitude of the cylinder oscillation, $\unicode[STIX]{x1D714}=2\unicode[STIX]{x03C0}f$ , $F_{L0}$ is the amplitude of the lift force and $\unicode[STIX]{x1D719}$ is the phase shift between the lift force and cylinder displacement.

The energy $W_{f}$ transferred from fluid to the cylinder motion for each cycle of oscillation is given by

(3.4) $$\begin{eqnarray}W_{f}=\int F\,\text{d}Y=\int _{t}^{t+T}F_{L0}\sin (\unicode[STIX]{x1D714}t^{\prime }+\unicode[STIX]{x1D719})\,\text{d}(Y_{0}\sin \unicode[STIX]{x1D714}t^{\prime })=\unicode[STIX]{x03C0}F_{L0}Y_{0}\sin \unicode[STIX]{x1D719}.\end{eqnarray}$$

Equation (3.4) implies that $W_{f}$ is directly proportional to $F_{L0}$ and $Y_{0}$ , and is positive when the lift force leads the displacement by $0^{\circ }<\unicode[STIX]{x1D719}<180^{\circ }$ . It should be noted that $W_{f}$ grows with $\unicode[STIX]{x1D719}$ when $\unicode[STIX]{x1D719}$ increases from $0^{\circ }$ to $90^{\circ }$ and then contracts for further increased $\unicode[STIX]{x1D719}$ . The energy dissipated by the structural damping is given by

(3.5) $$\begin{eqnarray}W_{d}=\int (-c{\dot{Y}})\,\text{d}Y=\int _{t}^{t+T}-c\unicode[STIX]{x1D714}Y_{0}\cos (\unicode[STIX]{x1D714}t^{\prime })\,\text{d}(Y_{0}\sin \unicode[STIX]{x1D714}t^{\prime })=-\unicode[STIX]{x03C0}c\unicode[STIX]{x1D714}Y_{0}^{2}.\end{eqnarray}$$

The component of force at the fundamental (body oscillation) frequency is in general considered to be dominant (Alam & Zhou Reference Alam and Zhou2007; Assi et al. Reference Assi, Bearman and Meneghini2010, Reference Assi, Bearman, Carmo, Meneghini, Sherwin and Willden2013). This is the most relevant component because the other frequency components do not yield a net contribution to the energy transfer between fluid and structure (Morse & Williamson Reference Morse and Williamson2009a ,Reference Morse and Williamson b ). This is also evident in figure 15(b–d).

The lift force can be split into two, viz.

(3.6) $$\begin{eqnarray}F_{L0}\sin (\unicode[STIX]{x1D714}t+\unicode[STIX]{x1D719})=F_{L0}\cos \unicode[STIX]{x1D719}\sin \unicode[STIX]{x1D714}t+F_{L0}\sin \unicode[STIX]{x1D719}\cos \unicode[STIX]{x1D714}t,\end{eqnarray}$$

where the first term on the right-hand side is in phase with the cylinder acceleration. Therefore, (3.1) can be written as

(3.7) $$\begin{eqnarray}\left(m+\frac{F_{L0}\cos \unicode[STIX]{x1D719}}{\unicode[STIX]{x1D714}^{2}Y_{0}}\right)\ddot{Y} +\left(c-\frac{F_{L0}\sin \unicode[STIX]{x1D719}}{\unicode[STIX]{x1D714}Y_{0}}\right){\dot{Y}}+kY=0.\end{eqnarray}$$

Equation (3.7) provides the added mass $m_{a}$ and added damping $c_{a}$ , namely,

(3.8a,b ) $$\begin{eqnarray}m_{a}=\frac{F_{L0}\cos \unicode[STIX]{x1D719}}{\unicode[STIX]{x1D714}^{2}Y_{0}}\quad \text{and}\quad c_{a}=-\frac{F_{L0}\sin \unicode[STIX]{x1D719}}{\unicode[STIX]{x1D714}Y_{0}}.\end{eqnarray}$$

Note that $m_{a}$ is the effective added mass caused by the lift force in phase with the cylinder acceleration, as opposed to the potential added mass.

Equation (3.7) can now be rewritten as

(3.9) $$\begin{eqnarray}\displaystyle & & \displaystyle \left(m-\lim _{n\rightarrow \infty }\frac{2}{nT(\unicode[STIX]{x1D714}^{2}Y_{0})^{2}}\int _{t}^{t+nT}F\,\ddot{Y} \,\text{d}t^{\prime }\right)\ddot{Y} \nonumber\\ \displaystyle & & \displaystyle \quad +\,\left(c-\lim _{n\rightarrow \infty }\frac{2}{nT(\unicode[STIX]{x1D714}Y_{0})^{2}}\int _{t}^{t+nT}F{\dot{Y}}\,\text{d}t^{\prime }\right){\dot{Y}}+kY=0.\end{eqnarray}$$

The integral terms associated with $\ddot{Y}$ and ${\dot{Y}}$ in (3.9) provide the means to estimate $m_{a}$ and $c_{a}$ , respectively, from experimental data, integrating over an integer number ( $n$ ) of oscillations. The added mass ratio $m_{a}^{\ast }$ and added damping ratio $\unicode[STIX]{x1D701}_{a}$ can then be estimated as

(3.10a ) $$\begin{eqnarray}\displaystyle & \displaystyle m_{a}^{\ast }=\frac{m_{a}}{m_{f}}=-\frac{8}{nT\unicode[STIX]{x1D70C}\unicode[STIX]{x03C0}D^{2}l(\unicode[STIX]{x1D714}^{2}Y_{0})^{2}}\int _{t}^{t+nT}F\ddot{Y} \,\text{d}t^{\prime }, & \displaystyle\end{eqnarray}$$

(3.10b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D701}_{a}=\frac{c_{a}}{c_{c}}=-\frac{1}{nTm(2\unicode[STIX]{x03C0}f_{n})(\unicode[STIX]{x1D714}Y_{0})^{2}}\int _{t}^{t+nT}F{\dot{Y}}\,\text{d}t^{\prime }\quad (\because c_{c}=4\unicode[STIX]{x03C0}mf_{n}), & \displaystyle\end{eqnarray}$$

$m_{f}$

$c_{c}$

$m_{a}^{\ast }$

$\unicode[STIX]{x1D701}_{a}$

$n$

$m_{a}^{\ast }$

$\unicode[STIX]{x1D701}_{a}$

$(n=1)$

$m_{a}^{\ast }\propto F\ddot{Y}$

$\unicode[STIX]{x1D701}_{a}\propto F{\dot{Y}}$

Figure 16. Variations in $Y_{0}/D$ , $\unicode[STIX]{x1D719}$ , $W_{f}$ , $\unicode[STIX]{x1D701}_{a}$ , $m_{a}^{\ast }$ and $C_{L\,rms}$ with $U_{r}$ .

3.4.3 Energy transfer during stable vibrations

At a given $U_{r}$ , the cylinder vibrates with almost unchanged amplitude $(\pm 4\,\%)$ . Thus $W_{f}$ , $m_{a}^{\ast }$ and $\unicode[STIX]{x1D701}_{a}$ can be calculated from (3.4) and (3.10), where $F_{L0}=\sqrt{2}\,F_{L\,rms}$ and $Y_{0}=\sqrt{2}\,Y_{rms}$ . The phase lag $\unicode[STIX]{x1D719}$ is calculated from the cross-correlations $R_{F_{L},Y}$ between $F_{L}$ and $Y$ signals obtained simultaneously, viz.

(3.11) $$\begin{eqnarray}R_{F_{L},Y}(\unicode[STIX]{x1D70F})=\frac{\overline{\{Y(t)\}\{F_{L}(t+\unicode[STIX]{x1D70F})\}}}{[\overline{\{Y(t)\}^{2}}]^{1/2}[\overline{\{F_{L}(t+\unicode[STIX]{x1D70F})\}^{2}}]^{1/2}},\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}$ is the time delay and $t$ is the time. This equation gives the cross-correlation coefficient as a function of $\unicode[STIX]{x1D70F}$ . The correlation coefficient is the maximum when the two signals are in phase. Yet, the two signals may not necessarily be in phase. Then a time delay given to one of the signals can make the two signals in phase, producing the maximum correlation coefficient. The time delay corresponding to the maximum correlation is scaled with a phase lag $\unicode[STIX]{x1D719}$ in angular degrees in one vibration cycle.

Figure 16 shows how $C_{L\,rms}~(=F_{L\,rms}/(0.5\unicode[STIX]{x1D70C}U_{\infty }^{2}lD))$ , $W_{f}$ , $m_{a}^{\ast }$ , $\unicode[STIX]{x1D701}_{a}$ and $\unicode[STIX]{x1D719}$ vary with $U_{r}$ or $Y_{0}/D$ . The structural damping ratio $\unicode[STIX]{x1D701}$ is $-$ 0.043. With increasing $U_{r}$ , $Y_{0}/D$ rises; so do $W_{f}$ and $C_{L\,rms}$ , with $\unicode[STIX]{x1D719}$ growing from ${\approx}131^{\circ }$ to $146^{\circ }$ . As noted before, an increase in $\unicode[STIX]{x1D719}$ beyond $90^{\circ }$ leads to a drop in $W_{f}$ . Thus, the rise in $Y_{0}/D$ is mainly caused by the increased $C_{L\,rms}$ , accompanied by an increase in $W_{f}$ . On the other hand, despite a rise in $Y_{0}/D$ with $U_{r}$ , $\unicode[STIX]{x1D701}_{a}$ changes little, hovering around the structural damping ratio $\unicode[STIX]{x1D701}$ (dashed line). The observation suggests that $|\unicode[STIX]{x1D701}|=|\unicode[STIX]{x1D701}_{a}|$ , where $\unicode[STIX]{x1D701}$ and $\unicode[STIX]{x1D701}_{a}$ are positive and negative, respectively. The small departure in the magnitude of $\unicode[STIX]{x1D701}_{a}$ from $\unicode[STIX]{x1D701}$ may originate from the assumption of sinusoidal $Y$ and $F_{L}$ variations (3.8). For a stable vibration, the energy transferred from fluid to structure should be equal to the energy dissipated by structural damping; that is, $W_{f}+W_{d}=0$ (equations (3.4) and (3.5)), i.e.  $\unicode[STIX]{x03C0}F_{L0}Y_{0}\sin \unicode[STIX]{x1D719}-\unicode[STIX]{x03C0}c\unicode[STIX]{x1D714}Y_{0}^{2}=0$ . Then, in view of (3.8), we obtain

(3.12a,b ) $$\begin{eqnarray}c=\frac{F_{L0}\sin \unicode[STIX]{x1D719}}{\unicode[STIX]{x1D714}Y_{0}}=-c_{a}\quad \text{or}\quad c+c_{a}=0.\end{eqnarray}$$

In other words, the added damping, invariant with $U_{r}$ , for a stable vibration simply compensates the structural damping so that the net damping is zero. Note that the energy dissipation due to the structural damping is proportional to $Y_{0}^{2}$ based on (3.5). The increase in $Y_{0}/D$ with $U_{r}$ thus requires more work done by the lift force. The $C_{L\,rms}$ is thus augmented accordingly to compensate the dissipation. Furthermore, as $\unicode[STIX]{x1D719}$ moves further away from $90^{\circ }$ with increasing $U_{r}$ , additional increase in $C_{L\,rms}$ is required to compensate its associated decrease in $W_{f}$ . The relationship between $Y_{0}$ , $F_{L0}$ and $\unicode[STIX]{x1D719}$ , which may cast light upon the above observation, can be obtained by equating (3.4) and (3.5), viz. $W_{f}+W_{d}=0$ . Then

(3.13) $$\begin{eqnarray}Y_{0}=\frac{F_{L0}\sin \unicode[STIX]{x1D719}}{c\unicode[STIX]{x1D714}}.\end{eqnarray}$$

That is, $Y_{0}$ is directly proportional to $F_{L0}$ .

Finally, the relationship between $m_{a}^{\ast }$ and $U_{r}$ may be analysed. Interestingly, $m_{a}^{\ast }$ decreases monotonically from approximately $-8$ to $-47$ as $U_{r}$ varies from 29.8 to 41.4 (figure 16 b).

3.5.1 Force and vibration characteristics in the initiation stage

The measured signals of $U_{r}$ , $Y/D$ , filtered $C_{L}$ (low-pass-filtered at a cut-off frequency of 20 Hz) are presented for the two cases in figure 18. In case I, $U_{r}$ rises continuously from 8.5 at $t=0~\text{s}$ (arbitrary) to 41.4 at $t=16.8~\text{s}$ and then remains unchanged (figure 18 a). The $Y/D$ remains at 0 until $t=8$  s when $U_{r}=28.3$ , and then starts to grow (figure 18 c). At $t\approx 9~\text{s}$ , $Y/D$ undergoes a rapid increase, reaching approximately 0.48 at $t=12~\text{s}$ , and then remains unchanged. It should be pointed out that, while $U_{r}$ changes from approximately 37 at $t=12~\text{s}$ to 41.4 at $t=16.8~\text{s}$ , $Y/D$ remains nearly constant, that is, $Y/D$ becomes saturated earlier than $U_{r}$ . As shown in figure 18(e), the $C_{L}$ amplitude grows very rapidly from 0.10 (the corresponding $A/D=0.02$ ) at $t=9~\text{s}$ to 1.48 ( $A/D=0.32$ ) at 9.8 s. After $t=9.8~\text{s}$ , the $C_{L}$ amplitude retreats slowly up to $t=13~\text{s}$ and then remains more or less constant. In case II, the cylinder is initially fixed with $U_{r}=41.4$ and then suddenly released at $t=1~\text{s}$ (figure 18 b). The $Y/D$ amplitude increases from approximately 0 to 0.48 at $t=1{-}2.9~\text{s}$ , and then remains constant (figure 18 d). The $C_{L}$ grows rapidly in amplitude up to $t=1.8~\text{s}$ , followed by a small contraction before reaching a rather stable state for $t>2.5~\text{s}$ (figure 18 f). In both cases, an interesting observation is that $C_{L}$ reaches its maximum magnitude in the transitional process before $Y/D$ reaches its stable amplitude. As such, we divide the transitional process into two: the initial transition where both $Y/D$ and $C_{L}$ grow in amplitude, and the late transition where $Y/D$ grows but $C_{L}$ slowly decreases in amplitude. The division point between the two transitions is characterized by the maximum amplitude of $C_{L}$ . The underlying physics behind the two transitions is different and will be discussed later. A close examination of the variation in $Y/D$ unveils that $Y/D$ grows rapidly in amplitude in the initial transition and mildly in the late transition, characterized by positive and negative curvatures, respectively. The maximum amplitude of $C_{L}$ occurs at the inflection point where the curvature is zero.

Figure 19. Time histories of $Y/D$ , $C_{L}$ and $f_{osc}/f_{n}$ in the transition process for (a,c,e) case I and (b,d,f) case II. The dashed lines in (a) and (b) enclose the $Y/D$ amplitude and those in (e) and (f) denote the trend of $f_{osc}/f_{n}$ . The dashed red lines in (c) and (d) are 20 Hz low-pass-filtered $C_{L}$ .

We next explore how the vibration is generated or how the vibration amplitude grows based on the signals of $Y/D$ , $C_{L}$ , low-pass-filtered $C_{L}$ at a cut-off frequency of 20 Hz and $f_{osc}/f_{n}$ in the transition stage of $t=8{-}11~\text{s}$ for case I and $t=1{-}3~\text{s}$ for case II (figure 19). The $f_{osc}/f_{n}$ in each cycle is simply estimated from the cylinder displacement of one peak to the next, i.e. one oscillation period ( $T$ ) and $f_{osc}=1/T$ . The sampling frequency of the $Y/D$ signal is 654 Hz, corresponding to an uncertainty in $f_{osc}$ of $\pm 0.0156f_{n}$ . The $Y/D$ signal is first 20 Hz low-pass-filtered before estimating $T$ so that the peaks may be identified with reasonable accuracy. In case I, $Y/D$ over $t=8{-}8.4~\text{s}$ experiences largely a sinusoidal variation with some minor high-frequency fluctuations due to the natural vortex shedding (refer to the inset of figure 19 a). However, $C_{L}$ fluctuates largely at the natural shedding frequency, which is much higher than the predominant frequency in $Y/D$ . The $C_{L}$ fluctuation at the cylinder oscillation frequency or the filtered $C_{L}$ (figure 19 c) is negligibly small at $t<8.4~\text{s}$ and starts to grow at $t\approx 8.4~\text{s}$ where the $C_{L}$ fluctuation at the shedding frequency is largest. At $t>8.4~\text{s}$ , the $C_{L}$ fluctuation decays at the shedding frequency but grows at the oscillation frequency. The observation is interesting, suggesting that the vibration may be initiated by natural vortex shedding. At $t=9.0{-}9.4~\text{s}$ where $Y/D$ grows rapidly in amplitude, the high-frequency fluctuation in $C_{L}$ still exists, along with the low-frequency fluctuation. At $t>9.4~\text{s}$ where $Y/D$ continues to grow in amplitude, the low-frequency fluctuation in $C_{L}$ becomes dominant, overwhelming the high-frequency fluctuation. As discussed before, the $C_{L}$ amplitude associated with the cylinder vibration frequency grows up to $t\approx 9.8~\text{s}$ and then dwindles, its maximum occurring at $t\approx 9.8~\text{s}$ . The $Y/D$ increases rapidly in amplitude over $t=8.4{-}9.8~\text{s}$ with a positive curvature (amplitude traced by the dashed line) and weakly with a negative curvature. Note that the curvature discussed here is based on the upper dashed line. Therefore, the durations of $t=8.4{-}9.8~\text{s}$ and $t>9.8~\text{s}$ are referred to as the initial and late transitions, respectively.

Figure 20. Power spectral density functions (a) $E_{Y}$ of $Y$ and (b) $E_{CL}$ of $C_{L}$ over different durations of time during the transition process for case I.

It is interesting to find out whether the cylinder oscillation frequency varies in the transition stage. As shown in figure 19(e), $f_{osc}/f_{n}$ jumps from 1.09 to approximately 1.3 at $t=8.4~\text{s}$ where the vibration is initiated, then declines up to $t=9.8~\text{s}$ . At $t>9.8~\text{s}$ , $f_{osc}/f_{n}$ varies little, slightly larger than 1. The duration of $t=8.0{-}8.4~\text{s}$ where $f_{osc}/f_{n}$ rises quickly is referred to as the pre-initial transition. Figure 20 presents the power spectra of $Y/D$ and $C_{L}$ over $t=8{-}8.4~\text{s}$ , 8.4–9.8 s and 9.8–11 s, corresponding to the pre-initial, initial and late transitions, respectively. The $Y/D$ spectrum over $t=8{-}8.4~\text{s}$ (figure 20 a) displays two pronounced peaks, corresponding to the cylinder oscillation and shedding frequencies, respectively. The magnitude of $f_{osc}/f_{n}$ is approximately 1.23. The $C_{L}$ spectrum over the same duration shows only one pronounced peak at the shedding frequency (figure 20 b). In the initial transition ( $t=8.4{-}9.8~\text{s}$ ), the $Y/D$ spectrum exhibits only one large peak at the cylinder oscillation frequency but the $C_{L}$ spectrum shows peaks at both cylinder oscillation and shedding frequencies, the peak at the oscillation frequency being smaller in amplitude than that at $t=8{-}8.4~\text{s}$ ( $f_{osc}/f_{n}=1.08$ ). On the other hand, in the late transition $(t=9.8{-}11~\text{s})$ , the $Y/D$ spectrum indicates that the cylinder oscillates at $f_{osc}/f_{n}\approx 1$ and the $C_{L}$ spectrum shows a tiny peak at the shedding frequency (figure 20 a,b). In summary, $f_{osc}/f_{n}$ increases in the pre-initial transition and reaches the maximum at the end of the pre-initial transition. The $f_{osc}/f_{n}$ value decreases in the initial transition and becomes almost constant in the late transition and afterwards. In the free vibration of a single cylinder, $f_{osc}$ in the lock-in range varies with $U_{r}$ at small $m^{\ast }\unicode[STIX]{x1D701}~({<}0.05)$ and low $m^{\ast }~({<}10)$ , and may deviate greatly from $f_{n}$ (Khalak & Williamson Reference Khalak and Williamson1999), while $f_{osc}/f_{n}\approx 1$ at large mass ratios, i.e.  $m^{\ast }>100$ (Govardhan & Williamson Reference Govardhan and Williamson2000). Govardhan & Williamson (Reference Govardhan and Williamson2000) identified the initial, upper and lower branches in the VE regime for a single cylinder at small $m^{\ast }\unicode[STIX]{x1D701}~(=0.0013)$ , and then obtained an expression for the vibration frequency in the lower branch, which takes into account the added mass ratio $m_{a}^{\ast }$ and is valid in the case of low $m^{\ast }\unicode[STIX]{x1D701}$ . Mittal & Kumar (Reference Mittal and Kumar2001) also noted the departure between $f_{osc}$ and $f_{n}$ for two tandem cylinders in cross-flow with $m^{\ast }=1.5$ and $m^{\ast }\unicode[STIX]{x1D701}=0.00051$ . However, the present large $m^{\ast }~(=275)$ and $m^{\ast }\unicode[STIX]{x1D701}~(=0.58)$ result in $f_{osc}/f_{n}\approx 1$ in a steady-state vibration (figure 19) for all $U_{r}$ examined. Therefore, the dependence of $f_{osc}/f_{n}$ on $U_{r}$ is not presented.

The same observation is made for case II, where the pre-initial, initial and late transitions correspond to $t=1{-}1.4~\text{s}$ , $t=1.4{-}1.8~\text{s}$ and $t>1.8~\text{s}$ , respectively. It may be concluded that natural vortex shedding is a predominant excitation source of the cylinder vibration in the pre-initial transition and initiates the vibration. However, both vortex shedding and fluctuating lift at $f_{osc}/f_{n}$ make important contributions to the cylinder vibration in the initial transition; yet it is the latter that is largely responsible for the vibration in the late transition. In the previous section, it has been demonstrated that in the stable vibration the gap shear layers interact vigorously with the vibrating downstream cylinder. It is now clear that in the transitional process the pre-initial transition is characterized by an interaction between natural vortex shedding and the cylinder; the gap shear layer joins the interaction between natural vortex shedding in the initial transition. However, the late transition is dominated by the interaction between the gap shear layer and the cylinder, which persists in the stable vibration.

3.5.2 Energy transfer and added mass

The roles of $\unicode[STIX]{x1D719}$ , $W_{f}$ and $m_{a}^{\ast }$ in the transition need to be clarified. The amplitude of the filtered $C_{L}$ is rather irregular and less sinusoidal in the first half $(t=8.4{-}9.0~\text{s})$ of the initial transition (figure 19 c) but becomes sinusoidal in the second half $(t>9.0~\text{s})$ . Similarly, in case II, the filtered $C_{L}$ is sinusoidal at $t>1.4~\text{s}$ when the initial transition starts. In the following discussion, the starting time is made at $t=9.0~\text{s}$ for case I and $t=1.4~\text{s}$ for case II. Figure 21 shows the signals of $Y_{0}/D$ ,  $C_{L0}$ , $\unicode[STIX]{x1D719}$ and $f_{osc}/f_{n}$ for the two cases, where $C_{L0}$ is the amplitude of the filtered $C_{L}$ in each cycle. In case I, $C_{L0}$ grows rapidly in the initial transition and approaches its maximum at $t=9.8~\text{s}$ at which $Y_{0}/D=0.32$ . In the same duration, $\unicode[STIX]{x1D719}$ increases from approximately $90^{\circ }$ to $134^{\circ }$ but $f_{osc}/f_{n}$ decreases monotonically from 1.09 to 1.01, both maintaining essentially a constant value at $t>11~\text{s}$ (figure 21 c,e). The changes in $\unicode[STIX]{x1D719}$ and $f_{osc}/f_{n}$ in the initial transition are largely connected to the gradual replacement of natural vortex shedding by the gap shear layers in the interaction with the downstream cylinder. In the late transition $(t=9.8{-}11~\text{s})$ , $C_{L0}$ retreats to approximately 1.1 with small variations in $\unicode[STIX]{x1D719}$ and $f_{osc}/f_{n}$ , whilst $Y_{0}/D$ is augmented to 0.48. A similar observation is made for case II (figure 21 b,d,f).

Figure 21. Time histories of $Y_{0}/D$ , $C_{L0}$ , $\unicode[STIX]{x1D719}$ and $f_{osc}/f_{n}$ for (a,c,e) case I and (b,d,f) case II.

The values of $W_{f}$ , $W_{d}$ and $W_{t}$  ( $=W_{f}+W_{d}$ ) vary at different instants and are calculated over one period for cases I and II (figure 22 a,b). The $W_{f}$ and $W_{d}$ values rise and decline, respectively, in the initial and late transitions and show no change after the late transition. On the other hand, $W_{t}$ grows rapidly in the initial transition, reaching a maximum before declining in the late transition. A value $W_{t}>0$ means $W_{f}>W_{d}$ , i.e. the energy transferred from the fluid to the cylinder is larger than the energy dissipated through structural damping. Therefore, $W_{t}>0$ implies that there is an excess of energy used to enlarge $Y_{0}/D$ . Naturally, the growth in $W_{t}$ in the initial transition is accompanied by a rise in $Y_{0}/D$ . In the late transition, the slowdown in the $Y_{0}/D$ growth is associated with a decline in $W_{t}$ . Once $W_{t}\approx 0$ , $Y_{0}/D$ becomes constant. Both $W_{f}$ and $W_{d}$ depend on $Y_{0}/D$ , as is evident in figure 22(c,d), where $-W_{d}$ is shown to make clear the difference between $W_{f}$ and $W_{d}$ . The difference between $W_{f}$ and $W_{d}$ grows from $Y_{0}/D=0$ to 0.32 and then tapers off, essentially collapsing together at $Y_{0}/D\approx 0.48$ .

Figure 22. Time histories of $W_{f}$ , $W_{d}$ , $W_{t}$ and $Y_{0}/D$ in each period during the transition for (a) case I and (b) case II. Variations in $W_{f}$ and $W_{d}$ with increase in $Y_{0}/D$ for (c) case I and (d) case II. Dashed lines are the least-squares curve fit to a second-order polynomial.

Finally, let us turn our attention to $m_{a}^{\ast }$ and $\unicode[STIX]{x1D701}_{a}$ . In case I, the vortex shedding frequency was detected in both $Y/D$ and $C_{L}$ in the pre-initial transition over $t=8{-}8.4~\text{s}$ (figure 20). Then, $m_{a}^{\ast }$ and $\unicode[STIX]{x1D701}_{a}$ are calculated based on the shedding frequency in the pre-initial transition but on the structural vibration for the rest of the transition. Interestingly, both $m_{a}^{\ast }$ and $\unicode[STIX]{x1D701}_{a}$ (figure 23) are initially positive at $t=8~\text{s}$ and then drop rapidly in the pre-initial transition. It seems plausible that the vibration is initiated by negative $m_{a}^{\ast }$ and $\unicode[STIX]{x1D701}_{a}$ , i.e. vortex shedding contributes to the initiation of the structural vibration. At $t=8.4~\text{s}{-}9.0~\text{s}$ , both $m_{a}^{\ast }$ and $\unicode[STIX]{x1D701}_{a}$ cannot be determined as discussed in § 3.5.1. At $t>9.0~\text{s}$ , both $m_{a}^{\ast }$ and $\unicode[STIX]{x1D701}_{a}$ are highly negative, showing a rapid rise from $-105$ to $-58$ and from $-0.26$ to $-0.07$ , respectively, in the initial transition and then a mild growth in the late transition. Their values go up in the stable vibration state to $m_{a}^{\ast }\approx -47$ and $\unicode[STIX]{x1D701}_{a}\approx -0.043$ , which is equal in magnitude to the structural damping, i.e.  $\unicode[STIX]{x1D701}_{a}+\unicode[STIX]{x1D701}=0$ (figure 23 c). A similar observation is made in case II (figure 23 b,d). It may be concluded that, during the growth of vibration, an effective damping ratio of the fluid–cylinder system is negative, i.e.  $\unicode[STIX]{x1D701}+\unicode[STIX]{x1D701}_{a}<0$ , resulting in an exponential growth in the vibration amplitude, while this effective damping ratio is zero, i.e.  $\unicode[STIX]{x1D701}+\unicode[STIX]{x1D701}_{a}=0$ , at the stable vibration state.

Figure 23. Time histories of $m_{a}^{\ast }$ and $\unicode[STIX]{x1D701}_{a}$ for (a,c) case I and (b,d) case II.

Following (3.4) and (3.5), a stable vibration for $c=0$ (undamped) leads to $\unicode[STIX]{x1D719}$ $=0^{\circ }$ or $180^{\circ }$ and hence $c_{a}=0$ (3.8). As such, for an undamped cylinder, $\unicode[STIX]{x1D719}$ is found to be $0^{\circ }$ for the initial branch (Govardhan & Williamson Reference Govardhan and Williamson2000; Prasanth & Mittal Reference Prasanth and Mittal2008) and $180^{\circ }$ for the upper and lower branches in the lock-in regime; $\unicode[STIX]{x1D719}$ is discontinuous between the initial and upper branches (Prasanth & Mittal Reference Prasanth and Mittal2008). For these values of $\unicode[STIX]{x1D719}$ , (3.4) yields $W_{f}=0$ , which is consistent with our intuition.