6.1 Flow structure

The flow structure is examined at $L/D=1.5$ and $U_{r}=11.3$ in order to understand the large galloping vibration of the upstream cylinder. Using the $Y^{\prime }$ signal as a reference for phase, we calculated the phase-averaged vorticity contours (figure 8 a–h), where each period $T$ of the cylinder oscillation is divided equally into eight instants/phases and each bin consists of 64 images. The PIV images were captured separately over $x^{\prime }/D=0{-}1.5$ and 1.5–7. Figure 8 shows one complete view window from $x^{\prime }/D=0{-}1.5$ and 1.5–7. In both cases, the $Y^{\prime }$ and $Y$ signals were also simultaneously captured, facilitating connection of the structural vibration and flow behaviours. The correspondence between the time instants/phases and $Y^{\prime }/D$ or $Y/D$ is given in figure 8(i). As the laser sheet was shining through from the bottom wall of the test section, the images show two blank strips (white colour) above the cylinders. The thick dashed arrows given above the downstream cylinder represent the flow directions predicted from the lower side of the snapshot at the opposite phase. For example, the qualitative feature of the shear layer behaviour on the upper side (white area) of the downstream cylinder at instants ( $b$ ) and ( $c$ ) can be viewed on the lower side at instants ( $f$ ) and ( $g$ ), respectively, when the upstream cylinder reaches the highest position for the former and the lowest for the latter.

Figure 8.

(a–h) Phase-averaged 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 positive and negative vorticities, respectively. The contour cutoff levels are $\pm 0.4$ and the increment is 0.4. (i) Variations in $Y/D$ and $Y^{\prime }/D$ where the time instants are specified by the dash-dotted lines corresponding to (a–h). $L/D=1.5$ , $U_{r}=11.3$ .

The value of $t/T=0$ (figure 8 a) refers to the instant when the upstream cylinder crosses the equilibrium position $Y^{\prime }/D=0$ at which the downstream cylinder is at the lowest position, both cylinders moving upwards. The upper shear layer around the upstream cylinder reattaches onto the front surface of the downstream cylinder (figure 8 a). The reattached flow runs largely over the front surface of the downstream cylinder. Two vortices B2 and A2 $^{\prime }$ are shed from the downstream cylinder, and B3 starts to emerge in the lower shear layer. As the upstream cylinder moves upwards (figure 8 b), the reattachment of the lower gap shear layer shifts upstream, feeding B3 with vorticity. The forward part of the reattached shear layer rolls into the gap of the cylinder, close to the base of the upstream cylinder. The upper gap shear layer detaches and generates vortex A3. When the upstream cylinder is at the top position (figure 8 c), the lower gap shear layer reattachment moves to the front surface, resulting in a strong rollup motion very close to the base of the upstream cylinder. The strong rollup motion may contribute to the large amplitude of vibration for the upstream cylinder. A small vortex B4 originates from the lower side reattached shear layer. Vortex A3 grows from instant ( $b$ ) to ( $c$ ). From instants ( $c$ ) and ( $d$ ), when the upstream cylinder moves downwards, the lower gap shear layer remains reattached. Vortex B3 grows in both size and vorticity concentration, originally acquiring its vorticity from the reattached shear layer and also acquiring vorticity from B4. As a result, B3 and B4 merge at instant ( $e$ ), forming B3 $^{\prime }$ ( $=$  B3 $+$ B4). Vortex A3, formed above the centreline, leads B3 $^{\prime }$ . As the upstream cylinder moves further downwards (figure 8 f), the upper shear layer reattaches on the upper-front surface of the downstream cylinder, and two new vortices A4 and B5 emerge, largely due to reattached and detached shear layers. Both A4 and B5 grow rapidly between instants ( $f$ ) and ( $g$ ) (figure 8 g). At instant ( $h$ ), the reattached shear layer results in a small vortex A5 that accompanies A4. Similarly to the merging of B3 and B4, both A4 and A5 merge and form a single vortex as A2 $^{\prime }$ in figure 8(a).

A total of six vortices (A3, B3, B4, A4, B5 and A5) are formed during one cycle of the cylinder oscillation, where B3 and A4 originate from the shear layers reattaching on the rear side ( $\unicode[STIX]{x1D703}=|90^{\circ }{-}120^{\circ }|$ , measured from the nominal front stagnation point) surface and merge with B4 and A5, respectively. The latter two vortices are generated during the reattachment shifting from the rear side to the front surface. Apparently, the wake downstream is characterized by four vortices, i.e. A3, B3 $^{\prime }$ ( $=$  B3  $+$  B4), A2 $^{\prime }$ ( $=$  A4  $+$  A5) and B5, which account for the second-harmonic lock-in, $f_{v2}/f_{n}=2.0$ . Interestingly, vortices A2 $^{\prime }$ and B3 $^{\prime }$ originating from the reattached shear layer gradually move towards the centreline and even cross it, showing up on the other side of the wake, while A3 and B5 stay on the same side, never crossing the centreline. It will be shown later that the vortices A3 and B5 that never cross the centreline persist further downstream, while A2 $^{\prime }$ and B3 $^{\prime }$ , that have crossed the centreline, are short-lived. It may be concluded that the reattachment and detachment of the gap shear layers contribute to the second-harmonic lock-in, and the violent vibration of the upstream cylinder may be connected to the reattached shear layer rollup motion very close to the base of the upstream cylinder.

6.2 Flow structure evolution

A summary sketch is given in figure 9 to illustrate the vortex evolution over $x/D=0{-}12$ . The cylinder position corresponds to instant ( $h$ ) and the vortices are placed in the wake based on their positions in figure 8(a–h). The streamwise separations between vortices B5 and B3 $^{\prime }$ and between B3 $^{\prime }$ and A3 are approximately $2.6D$ and $1.0D$ , respectively, as estimated from figure 8(h). Similarly, the streamwise distances between A3 and A2 $^{\prime }$ , between A2 $^{\prime }$ and B2, and between B2 and B1 are $2.8D$ , $1.0D$ and $2.8D$ , respectively, obtained from figures 8(e), 8(c) and 8(a) in view of the observation that the convection velocity of vortices changes little beyond $x/D=5.5$ (Lin & Hsieh Reference Lin and Hsieh2003; Alam & Zhou Reference Alam and Zhou2007). The vortices A3 and B3 $^{\prime }$ , close to each other, may be considered to be one pair. So are A2 $^{\prime }$ and B2. It is noted that the signs of the vortices in the wake follow the pattern of positive (B2) $\rightarrow$ negative (A2 $^{\prime }$ ) $\rightarrow$ negative (A3) $\rightarrow$ positive (B3 $^{\prime }$ ). The orientation of the vortices is similar to the 2P wake mode for the vortex-induced vibration of a single cylinder (Williamson & Roshko Reference Williamson and Roshko1988). The difference is that, for the single cylinder, the vortex stretching and splitting process appears to be fundamental to the 2P wake formation mode (Govardhan & Williamson Reference Govardhan and Williamson2000), where one vortex stretches and then splits into two vortices, one staying on the same side while the other crosses the centreline. The latter is called an inverted vortex. The inverted vortices were characterized by their relative weakness and decayed more rapidly than those that never crossed the centreline. On the other hand, for the two-cylinder case, there is no evidence for vortex stretching and splitting during the vortex evolution. Rather, vortex merging takes place, producing B3 $^{\prime }$ and A2 $^{\prime }$ , and the 2P mode is produced via the evolution of six vortices.

Figure 9.

Sketch of the vortex street behind the cylinders, following figure 8. P stands for a vortex pair. The red dashed-dotted line encircles the vortices shed in one oscillation cycle.

6.3 Pressure distribution around vibrating cylinders

The phase-averaged pressure coefficient $\langle C_{P}\rangle$ distributions around both cylinders are superimposed with the phase-averaged vorticity structures to explore the vibration mechanism, as shown in figure 10 (for instants i–v). Figure 11(a,b) shows the superimposed plots (50 cycles) of the cylinder displacement and lift force coefficient ( $C_{Lu}$ for the upstream cylinder and $C_{Ld}$ for the downstream) in one oscillation cycle as a function of $t/T$ .

Figure 10.

Phase-averaged vorticity contours and pressure coefficient $\langle C_{P}\rangle$ around cylinders in one half-cycle of the cylinder oscillation for instants (i)–(v). The contour cutoff levels are $\pm 0.4$ and the increment is 0.4. $L/D=1.5$ . $A_{u}^{\ast }=0.3$ and $A_{d}^{\ast }=0.07$ .

Figure 11.

Superimposed plots (50 cycles) of displacement and lift force variations. (a) $Y^{\prime }/D$ and $C_{Lu}$ , and (b) $Y/D$ and $C_{Ld}$ . Power spectral density functions (c) $E_{Y^{\prime }}$ and $E_{C_{Lu}}$ , and (d) $E_{Y}$ and $E_{C_{Ld}}$ . The instants marked above the figures represent the instants of the snapshots in figure 10. $L/D=1.5$ . $A_{u}^{\ast }=0.3$ and $A_{d}^{\ast }=0.07$ .

At instant (i) in figure 10, the upper shear layer separated from the upstream cylinder reattaches onto the front-side surface of the downstream cylinder, leading to a maximum $\langle C_{P}\rangle \,(=0.175)$ at $\unicode[STIX]{x1D703}=56.25^{\circ }$ . Part of the reattached flow passes over the upper-side surface of the downstream cylinder and the other part impinges upon the front surface, both causing negative pressure on the surfaces in contact. As a result, $C_{Ld}$ on the downstream cylinder is above 0 ( ${\approx}0{-}0.04$ , figure 11 b). For the upstream cylinder, the stagnation point occurs at $\unicode[STIX]{x1D703}=0^{\circ }$ , where $\langle C_{P}\rangle \approx 1$ . Due to the rollup of the upper shear layer in the gap, the negative pressure around the upper-rear surface of the upstream cylinder is associated with a larger magnitude than that around the lower-rear surface, resulting in a positive $C_{Lu}$ (figure 11 a). Shear layer separation or growth is accompanied by a higher velocity and a lower pressure (Nishimura & Taniike Reference Nishimura and Taniike2001; Alam Reference Alam2016). Similarly, the rollup of the upper gap shear layer may cause more negative pressure on the upstream cylinder than that of the lower shear layer. As the upstream cylinder moves upwards at instant (ii), the lower gap shear layer also rolls up with more negative pressure on the lower surface and less negative pressure on the upper surface. The $C_{Lu}$ thus decreases, though remaining above 0 (figure 11 a). When the upstream cylinder reaches the top and then moves down (instants iii and iv), the rollup of the lower gap shear layer, moving closer to the base of the upstream cylinder, generates pronouncedly negative pressure on the lower-rear surface. The value of $C_{Lu}$ is around 0 at instants (iii) and (iv). The pressure on the lower-rear surface becomes even more negative at instant (v) due to the strong rollup of the lower gap shear layer, along with less negative pressure on the upper-rear surface, giving rise to a highly negative $C_{Lu}$ (figure 11 a). On the whole, the rollup of the gap shear layers, largely affecting the pressure distribution on the side and rear surfaces of the upstream cylinder, contributes to the large vibration of the upstream cylinder. As the gap shear layer rollup near the base region of the upstream cylinder is connected to the upstream cylinder oscillation, not to vortex shedding behind the downstream cylinder, the upstream cylinder vibration does not stop or alter when the downstream cylinder is stationary (figure 7 a).

The phase-averaged velocity vectors at instants (i) $t=0$ and (v) $t=T/2$ of figure 10 are presented in figure 12 to show clearly how the flow behaves in the gap. As shown at instant (i) in figure 10 at $t=0$ , the upper shear layer separated from the upstream cylinder reattaches onto the front surface of the downstream cylinder. Part of the reattached flow passes over the upper-side surface of the downstream cylinder and the other part impinges upon the front surface (figure 10, instant i), with a strong flow running downwards over the front surface (figure 12 a). A clockwise recirculation thus forms between the cylinders. The impinging flow corresponds to the negative pressure on the front surface (figures 10, instant i, 12 a). Similarly, at $t=T/2$ , where the lower shear layer is reattached (figure 10, instant v), the recirculation bubble is anticlockwise (figure 12 b), causing the negative pressure on the front surface (figure 10, instant v).

Figure 12.

Phase-averaged velocity vectors at instants (a) instant (i) $t=0$ and (b) instant (v) $t=T/2$ in figure 10. $L/D=1.5$ . $A_{u}^{\ast }=0.3$ and $A_{d}^{\ast }=0.07$ .

The pressure pattern of the downstream cylinder seems more complicated. Due to the reattachment of the lower gap shear layer on the lower surface of the downstream cylinder, the maximum pressure occurs at $\unicode[STIX]{x1D703}=303.8^{\circ }$ , where $\langle C_{P}\rangle =-0.13$ , 0.48, and 0.13 for instants (iii), (iv) and (v), respectively. The value of $C_{Ld}$ is highly negative at instant (iii) but becomes highly positive at instant (iv), dropping slightly at instant (v) (figure 11 b). The complicated pressure pattern on the downstream cylinder arises from the interactions of the downstream cylinder with the gap shear layers as well as its own vortex shedding. The power spectral density functions $E_{Y^{\prime }}$ of $Y^{\prime }$ and $E_{C_{Lu}}$ of $C_{Lu}$ are presented in figures 11(c) and 11(d) for the upstream and downstream cylinders, respectively. Both $E_{Y^{\prime }}$ and $E_{Y}$ are characterized by one predominant peak at $f_{n}$ , while $C_{Lu}$ displays one more pronounced peak at $3f_{n}$ (figure 11 c). On the other hand, the pronounced peak at $3f_{n}$ in $E_{C_{Ld}}$ of $C_{Ld}$ on the downstream cylinder (figure 11 d) overwhelms that at $f_{n}$ . Obviously, the peak at $f_{n}$ is linked to the cylinder oscillation at $f_{n}$ and that at $3f_{n}$ is connected to the shedding of six vortices in one cycle of the cylinder oscillation, as explained in figure 8. Overall, $C_{Lu}$ is dominated by the oscillation frequency ( $=f_{n}$ ) (figure 11 a,c) that is connected with the rollup of the shear layers, though the fluctuation at $3f_{n}$ is obviously captured in $C_{Ld}$ (figure 11 b,d), which is linked to the shedding of six vortices in one cycle of the cylinder oscillation. Interestingly, although the maximum value of $C_{Ld}$ ( ${\approx}$  0.8) is larger than that of $C_{Lu}$ ( ${\approx}$  0.6), the amplitude of $Y^{\prime }/D$ ( ${\approx}0.3$ ) is much bigger than the amplitude of $Y/D$ ( ${\approx}0.07$ ) (figure 11 a,b). The large vibration of the upstream cylinder will be elaborated in the next section in terms of energy transfer from the flow to the upstream cylinder.

6.4 Work done

Whether a vibration is sustainable depends on the energy transfer between the flow and cylinder, which is subsequently linked to the way vortices are separated from the cylinder. It is important to understand the work done on the upstream cylinder by $C_{Lu}$ during its vibration. Figure 13(a,b) presents the temporal evolution of instantaneous work done for the upstream cylinder, $W_{fi}\,(=C_{Lu}\times d(Y^{\prime }/D))$ in one cycle of oscillation, where subscripts 1 and 2 denote $C_{Lu}$ containing the oscillation frequency and vortex shedding frequency components, respectively. More specifically, $C_{Lu,1}$ is the 20 Hz low-pass-filtered $C_{Lu}$ , and $C_{Lu,2}=C_{Lu}-C_{Lu,1}$ . Note that $C_{Lu}$ is first low-pass-filtered at $4f_{n}$ , which is larger than the vortex shedding frequency to exclude the high-frequency noise. As discussed before, the violent vibration of the upstream cylinder may be connected to the reattached shear layer rollup near the base of the upstream cylinder, the rollup frequency being the oscillation frequency. Thus, $W_{fi,1}\,(=C_{Lu,1}\times d(Y^{\prime }/D))$ can be considered as the work done by the gap shear layers and $W_{fi,2}\,(=C_{Lu,2}\times d(Y^{\prime }/D))$ as the work done by vortices shed during the cylinder oscillation. Positive and negative work done indicate the flow acting to excite and dampen the vibration, respectively. In other words, positive work done means the cylinder receives energy from the flow, while the energy transfer is opposite given negative work done.

Figure 13.

(a,b) Time histories of $Y^{\prime }/D$ , $C_{Lu}$ and $W_{fi}$ in one cycle of the upstream cylinder vibration. (c) Variations in $\sum W_{fi}$ , $\sum W_{fi,1}$ , and $\sum W_{fi,2}$ in fifteen cycles. The subscripts 1 and 2 denote the $C_{Lu}$ or $W_{fi,}$ containing the oscillation frequency and vortex shedding frequency components, respectively. $L/D=1.5$ . $A_{u}^{\ast }=0.3$ and $A_{d}^{\ast }=0.07$ .

As presented in figure 13(a), $C_{Lu,1}$ increases slightly, reaching a maximum value at $t/T=0.04$ , and then decreases to approximately 0.07 as the upstream cylinder increases from $Y^{\prime }/D=0$ to 0.3 (maximum). The positive $W_{fi,1}$ increases first for $t/T=0{-}0.02$ $(Y^{\prime }/D=0{-}0.02)$ and then decreases until $t/T=0.25$ (process I, marked at the top of figure 13 a). The cylinder absorbs the largest amount of energy at $t/T=0.02$ , where the reattachment, rollup and separation of the upper shear layer in the gap take place to generate the positive $C_{Lu,1}$ (figures 10 instant i, 13 a). With an increase in $Y^{\prime }/D$ , the cylinder receives less energy because the upper gap shear layer is detached from the downstream cylinder, resulting in less circulation in the gap (figure 10ii, iii) and decreasing $C_{Lu,1}$ (figure 13 a). When the cylinder moves downwards from the top to $Y^{\prime }/D=0.28$ , $W_{fi,1}$ becomes negative, just below zero, for a short duration (process II). Then, $W_{fi,1}$ becomes positive again and increases as the cylinder continues drop downwards to the centreline (process III). Obviously, the reattachment, rollup and separation of the lower shear layer take place now in the gap, generating negative $C_{Lu,1}$ to drive the cylinder in the same direction. As such, $W_{fi,1}$ increases until the cylinder reaches its centreline $(Y^{\prime }/D=0)$ position. The work done or $W_{fi,1}$ is summarized in figure 14(a). The cylinder receives energy from the flow when moving upwards to the maximum $Y^{\prime }/D$ (process I). As the cylinder moves from the top to $Y^{\prime }/D=0.28$ , the work done is negative, slightly below zero. As the cylinder moves downwards from $Y^{\prime }/D=0.28$ to $Y^{\prime }/D=0$ , $W_{fi,1}$ becomes positive again. The observation suggests that the rollup of the gap shear layers around the rear surface of the upstream cylinder contributes to the positive work done from the flow to the cylinder for almost the entire process, supporting the vibration. The upstream cylinder vibration may also be affected by vortex shedding from the downstream cylinder as the two cylinders are closely separated. The time-averaged value of $W_{fi,1}$ is positive, indicating that the vibration may be sustained with the cylinder absorbing energy at $f_{n}$ .

Figure 14.

Sketch for the signs of work done in one half-cycle of the upstream cylinder oscillation (a) $W_{fi,1}$ , (b) $W_{fi,2}$ .  $L/D=1.5$ . $A_{u}^{\ast }=0.3$ and $A_{d}^{\ast }=0.07$ .

As to $W_{fi,2}$ , four processes can be identified for one half-cycle of oscillation, as indicated at the top of figure 13(b). There are approximately three oscillations of $C_{Lu,2}$ in one period of the cylinder vibration, suggesting the fluctuation in $C_{Lu,2}$ is connected to the six vortices shed in one vibration cycle, as described above. Naturally, the reattachment may make the fluctuating period of $C_{Lu,2}$ non-uniform. The largest energy absorbed by the cylinder occurs at $Y^{\prime }/D=0$ , where the shedding of vortex A2 $^{\prime }$ makes $C_{Lu,2}$ positive (figure 10, instant i). When the cylinder moves upwards to $Y^{\prime }/D=0.14$ , both $C_{Lu,2}$ and $W_{fi,2}$ drop to 0 (process I $^{\prime }$ ). The growth of vortex B3 (figure 10, instants i and ii) is linked to the decay of $C_{Lu,2}$ as well as $W_{fi,2}$ . With the cylinder continuing to move to the top and then goes down slightly to $Y^{\prime }/D=0.29$ , the negative $C_{Lu,2}$ dwindles and then augments; so does $W_{fi,2}$ (process II $^{\prime }$ ). The growth of B3 generates negative lift resisting the upward motion (figure 10, instants ii and iii), which is responsible for the further decrease in $W_{fi,2}$ in the first half of process II $^{\prime }$ . On the other hand, $W_{fi,2}$ increases in the second half of process II $^{\prime }$ because of the growth and shedding of A3 (figure 10, instant iii). The $C_{Lu,2}$ that arises to its maximum then diminishes, while the negative $W_{fi,2}$ decreases and then increases when the cylinder shifts to $Y^{\prime }/D=0.14$ (process III $^{\prime }$ ). The process involves the growth and amalgamation of B3 and B4. In process IV $^{\prime }$ , where $C_{Lu,2}$ becomes negative and drives the cylinder in the same direction, $W_{fi,2}$ goes up rapidly from 0, and the shedding of B3 $^{\prime }$ contributes to the work done. The work done or $W_{fi,2}$ is summarized in figure 14(b). The $W_{fi,2}$ is positive when the cylinder is near the centreline but becomes negative when the cylinder is away from the cylinder. In other words, positive $W_{fi,2}$ is associated with vortices A2 $^{\prime }$ and B3 $^{\prime }$ , while the most negative $W_{fi,2}$ results from A3 with the cylinder moving from the top (figures 10, instants ii and iv; 13 b, 14 b). This explains why A3 persists downstream and A2 $^{\prime }$ and B3 $^{\prime }$ diminish more rapidly than A3 (figure 9).

The cumulative work $\sum W_{fi}\,(=\sum C_{Lu}\times d(Y^{\prime }/D))$ in fifteen cycles is presented in figure 13(c) to understand how $W_{fi,1}$ and $W_{fi,2}$ contribute to sustaining of the cylinder vibration. Obviously, $\sum W_{fi,1}$ for fifteen cycles is approximately 2.3 at $t/T=15$ , essentially the same as $\sum W_{fi}$ since $\sum W_{fi,2}\approx 0$ . The observation indicates that energy transferred from the flow to the cylinder to compensate the energy dissipation by fluid–structure system damping is largely contributed from $\sum W_{fi,1}$ , rather than $\sum W_{fi,2}$ .

Figure 15.

(a–h) Phase-averaged vorticity $\unicode[STIX]{x1D714}_{z}^{\ast }\,(=\unicode[STIX]{x1D714}_{z}D/U_{\infty })$ contours in one oscillation cycle. Solid and dotted contour lines stand for positive and negative vorticities. The contour cutoff levels are $\pm 0.4$ and the increment is 0.4. (i) Variations in $Y/D$ and $Y^{\prime }/D$ , where the time instants are specified by the dash-dotted lines corresponding to (a–h). $L/D=2.5$ , $U_{r}=35$ .

7.1 Flow structure

The phase-averaged vorticity contours are presented in figure 15 for one period of structural vibration at $L/D=2.5$ and $U_{r}=35$ with a view to understanding why $A_{u}^{\ast }<A_{d}^{\ast }$ in regime III. The phase-averaged process is done using the $Y$ signal as a reference. At $t=0$ ( $Y/D=0$ and $Y^{\prime }/D=0.04$ ), both cylinders are close to the centreline. The two gap shear layers reattach on the downstream cylinder (figure 15 a). The lateral separation of the upper shear layer from the wake centreline $(y^{\prime }/D=0)$ is, however, smaller than that of the lower shear layer. Part of the lower reattached shear layer starts its rollup on the front surface of the downstream cylinder. Vortex shedding from the downstream cylinder occurs largely from the upper side, vortex A2 appearing larger in size than vortex B2. As the downstream cylinder moves upwards (figure 15 b), the upper shear layer rolls up on the front surface, and a new vortex A3 originates from the shear layer over the upper side of the downstream cylinder, not from the gap shear layer and not directly interacting with the gap shear layer or gap vortices. Due to the upward movement of the cylinder, a shear layer with positive (anticlockwise, red colour) vorticity emerges on the front surface and joins the lower shear layer from the upstream cylinder, both contributing to the generation of vortex B2 (figure 15 b). When $Y/D$ is large enough (figure 15 c), the upper gap shear layer swerves downwards without reattachment and interacts with vortex B3 generated by the lower shear layer of the downstream cylinder. Both gap shear layers contribute vorticity to B3. B3 is thus not so well structured as A3. Another vortex A4 grows from the upper shear layer, again not directly interacting with the gap shear layers. Though each cylinder has two distinct shear layers, the shear layers around the downstream cylinder roll up very near the cylinder, while those around the upstream cylinder reattach on, detach from, or roll up, depending on $Y/D$ , around the downstream cylinder, which leads to the shear layer swerving from one side to the other. The downward movement of the cylinder causes rollup of the upper gap shear layer very close to the downstream cylinder (figure 15 d). When the cylinder moves back to $Y/D=0$ (figure 15 e), both gap shear layers reattach and part of the upper shear layer rolls up on the front surface. The ensuing flow structure is a mirror image of that in figure 15(a). Similarly, the flow structures in figure 15(f–h) are mirror images of those in figure 15(b–d), respectively, which enable us to recover the vorticity structure on the upper side (masked area) of the downstream cylinder.

The vortices shed from the downstream cylinder are inclined downwards at instants ( $a$ ) and ( $b$ ) when the cylinder moves upwards. Vortex shedding appears almost symmetric at instant ( $c$ ), at which the cylinder velocity is zero (maximum $Y/D$ ). As the cylinder moves downwards from instant ( $c$ ) to ( $g$ ), vortex shedding from the downstream cylinder is deflected upwards. The maximum deflection occurs at instants ( $a$ ) and ( $e$ ) when the cylinder velocity is maximum. A total of eleven vortices are formed in one cycle of the cylinder oscillation (A2–A7 and B2–B6), accounting for $f_{v2}/f_{n}\approx 5.5$ (figure 5 d). There are two types of vortex generated in the wake: interacting and non-interacting with the gap shear layers. The latter type includes A3, A4, B5 and B6, which are well structured and decay slowly, formed when the cylinder is away from the centreline (instants $b$ , $c$ , $f$ – $h$ ). The other vortices (A2, A5, A6, A7, B2, B3 and B4) interact with the gap shear layers and decay rather rapidly. They receive vorticity from the gap shear layers during their generation.

It is also noted that the two gap shear layers pass over the lower and upper sides of the downstream cylinder at the maximum and minimum $Y/D$ (figure 15 c,g), respectively. Obviously, the alternating reattachment and switching of the two gap shear layers from one side to the other follow the cylinder oscillation, and the switching frequency is synchronized with the cylinder oscillation. The rollup and impingement of the gap shear layers occur around the downstream cylinder, away from the upstream cylinder, which explains why $A_{d}^{\ast }>A_{u}^{\ast }$ . As the switch of the gap shear layers is connected to the downstream cylinder oscillation, the upstream cylinder vibration stops when the downstream cylinder is stationary (figure 7 c).

7.2 Pressure distribution around vibrating cylinders

Figure 16 presents the distributions of the phase-averaged pressure coefficient $\langle C_{P}\rangle$ around both cylinders, along with phase-averaged vorticity contours for five instants (i) to (v). The $\langle C_{P}\rangle$ distribution on the surface of the upstream cylinder does not change much from instant (i) to (v). The $\langle C_{P}\rangle$ of the upstream cylinder decreases from the maximum (1.0) at $\unicode[STIX]{x1D703}=0^{\circ }$ (stagnation point) to the minimum $({\approx}-1)$ at $\unicode[STIX]{x1D703}=56.25^{\circ }$ , but changes little on the rear surface. The distribution is similar to the time-averaged pressure distribution on the upstream cylinder at $L/D=2.2$ and $Re=6.5\times 10^{4}$ for two tandem fixed cylinders (Alam et al. Reference Alam, Moriya, Takai and Sakamoto2003). As shown in figure 17(a), $C_{Lu}$ varies in a small range (from $-0.18$ to 0.18) in line with the small change in the $\langle C_{P}\rangle$ distribution at different instants (figure 16). The value of $Y^{\prime }/D$ is periodic $(f_{n}=1/T)$ ; yet $C_{Lu}$ is obviously chaotic, containing multiple frequencies (figure 17 a). On the other hand, $\langle C_{P}\rangle$ around the downstream cylinder surface at instant (i) is negative; its maximum $(\langle C_{P}\rangle \approx -0.07)$ occurring at $\unicode[STIX]{x1D703}\approx 67.5^{\circ }$ . The rollup of the lower reattached shear layer on the front surface of the downstream cylinder gives rise to a negative pressure at $\unicode[STIX]{x1D703}=-60^{\circ }{-}0^{\circ }$ and $\langle C_{P}\rangle \approx -0.60{-}-0.87$ (instant i), leading to $C_{Ld}$ slightly lower than zero (figure 17 b). At instant (ii) where the cylinder is away from the centreline, the negative pressure on the front surface results from the rollup of the upper gap shear layer and from the boundary layer generated with positive vorticity. The boundary layer on the upper side, its separation and following strong rollup motion produce a pronouncedly negative pressure on the side surface, with a maximum magnitude of 2.02 at $\unicode[STIX]{x1D703}=90^{\circ }$ . The corresponding positive $C_{Ld}$ is maximum. The $C_{Ld}$ initially decreases from instant (i) to (ii), followed by a rapid increase to the maximum. While the mild decrease is connected to the growth in B2, the rapid increase is attributed to the upper boundary layer and strong rollup motion of A3 near the cylinder surface. From instant (ii) to (iii), the pressure pattern changes markedly, the pronouncedly negative pressure shifting from the upper side to the lower (minimum $\langle C_{P}\rangle \approx -2.05$ at $\unicode[STIX]{x1D703}\approx 303.8^{\circ }$ ). Two factors account for this change. One is the upper gap shear layer of positive vorticity flows over the lower side, and the other is the formation of the boundary layer with strong vorticity on the lower side (instant iii). Couder & Basdevant (Reference Couder and Basdevant1986) demonstrate experimentally and numerically that the convective behaviours of two oppositely signed vortices may give rise to an increase in flow velocity at their interface. It is thus expected that the increased velocity over the lower side also contributes to the negative pressure. Note that instant (iii) corresponds to a kink (small peak) between the maximum and minimum $C_{Ld}$ (figure 17 b). The kink results from the formation of vortex A4 on the upper side. With the cylinder moving towards the centreline from instant (iii) to (iv), the maximum magnitude of the negative pressure region shifts from the lower side surface to the front surface, essentially ascribed to the change in the upper gap shear layer rollup (instants iii–v). Interestingly, it is the interaction between the gap shear layers and the downstream cylinder that is the primary cause of the change in the pressure distribution over the front and side surfaces during the cylinder motion.

Figure 16.

Phase-averaged vorticity contours and pressure coefficient $\langle C_{P}\rangle$ around cylinders in one half-cycle of the cylinder oscillation (instants i–v). The contour cutoff levels are $\pm 0.4$ and the increment is 0.4. $L/D=2.5$ , $A_{d}^{\ast }=0.6$ .

Figure 17.

Superimposed plots (50 cycles) of displacement and lift force variations. (a) $Y^{\prime }/D$ and $C_{Lu}$ , and (b) $Y/D$ and $C_{Ld}$ . Power spectral density functions (c) $E_{Y^{\prime }}$ and $E_{C_{Lu}}$ , and (d) $E_{Y}$ and $E_{C_{Ld}}$ . The instants marked above the figures represent the instants of the snapshots in figure 16. $L/D=2.5$ , $A_{d}^{\ast }=0.6$ .

At $L/D=2.5$ (figure 17 c,d), $E_{Y^{\prime }}$ clearly displays one peak at $f_{n,}$ , and the peak at $f_{n}$ in $E_{C_{Lu}}$ is not so pronounced as at $L/D=1.5$ (figure 11 c), suggesting that the small-amplitude vibration of the upstream cylinder is largely induced by the vibration of the downstream cylinder. As such, when the downstream cylinder is fixed, the vibration of the upstream cylinder, albeit small, is suppressed (figure 7 c). The $E_{C_{Ld}}$ shows two predominant peaks at $f_{n}$ and $3f_{n,}$ , respectively (figure 17 d), the former is more pronounced than the latter. There are also other superharmonic peaks in both $E_{C_{Lu}}$ and $E_{C_{Ld}}$ , which are perhaps linked to the strong interactions between the shear layers, separated from the upstream cylinder, and the downstream cylinder.

7.3 Work done

Figure 18(a,b) presents the temporal evolution of instantaneous work of the downstream cylinder, $W_{fi}\,(=C_{Ld}\times d(Y/D))$ in one cycle of oscillation, where subscripts 1 and 2 denote the components of $C_{Ld}$ associated with the cylinder oscillation and vortex shedding frequencies, respectively. As in regime I (§ 6.4), $C_{Ld,1}$ is 20 Hz low-pass-filtered $C_{Ld}$ and then $C_{Ld,2}=C_{Ld}{-}C_{Ld,1}$ . Note that $C_{Ld}$ is first low-pass filtered at $6f_{n}$ , which is higher than the vortex shedding frequency. As noted earlier, the gap shear layers switch from one side to the other at a frequency equal to the cylinder vibration frequency. Therefore, $W_{fi,1}\,(=C_{Ld,1}\times d(Y/D))$ can be considered as the work done by the gap shear layer switch and $W_{fi,2}\,(=C_{Ld,2}\times d(Y/D))$ as the work done by vortices shed during the cylinder oscillation.

Figure 18.

Time histories of $Y/D$ , $C_{Ld}$ and $W_{fi}$ in one cycle: (a) $C_{Ld,1}$ and $W_{fi,1}$ and (b) $C_{Ld,2}$ and $W_{fi,2}$ . (c) Variations in $\sum W_{fi}$ , $\sum W_{fi,1}$ and $\sum W_{fi,2}$ in fifteen cycles. $L/D=2.5$ , $A_{d}^{\ast }=0.6$ .

As shown in figure 18(a), $C_{Ld,1}$ decreases to zero as the cylinder moves upwards from $Y/D=0$ to 0.21, resulting in a positive but decreasing $W_{fi,1}$ (process I, marked at the top of figure 18 a). As the cylinder continues its upward movement till the top position, where $d(Y/D)=0$ , $W_{fi,1}$ becomes negative (process II). 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 III). Evidently, there is hysteresis in the half-cycle of oscillation due to a phase lag between $C_{Ld,1}$ and $Y/D$ , which results from the interaction between the gap shear layers and the downstream cylinder, as observed in the temporal evolution of the flow structure in figures 15 and 16. The work done is summarized in figure 19(a). While the cylinder moves upwards from $Y/D=0$ to $Y/D=0.21$ , the work done is positive and the cylinder absorbs energy from the flow (process I) where the shear layer reattachment switches from the lower side to the upper side (figure 16, instants i and ii). From $Y/D=0.21$ to its maximum, the work done is negative and energy is transferred from the vibrating cylinder to the flow, due to the separation of the upper gap shear layer from the downstream cylinder (figure 16, instants ii and iii). On the other hand, when moving from the highest position to $Y/D=0$ , the cylinder absorbs energy from the flow for the entire process III, and the maximum energy transfer occurs at $Y/D=0.32$ . Interestingly, in process III, the upper gap shear layer reattaches and the reattachment point shifts from the lower side to the upper side, resulting in energy transfer from the flow to the cylinder (figure 16, instants iii–v). The observation suggests that the shear layer reattachment, detachment and switch act to sustain the cylinder vibration.

Figure 19.

Sketch for the signs of work done in one half-cycle of the downstream cylinder oscillation. (a) $W_{fi,1}$ and (b) $W_{fi,2}$ .  $L/D=2.5$ , $A_{d}^{\ast }=0.6$ .

Consider the work done by vortices (figure 18 b). As $C_{Ld,2}$ decreases due to the growth in vortex B2, the negative $W_{fi,2}$ experiences a small decrease (process I $^{\prime }$ ). Afterwards, $C_{Ld,2}$ rises rapidly, which is connected to the strong rollup motion of vortex A3 near the cylinder surface. As a result, $W_{fi,2}$ goes up rapidly from a negative value to its maximum (process II $^{\prime }$ ). Both $C_{Ld,2}$ and $W_{fi,2}$ decline promptly (process III $^{\prime }$ ) due to the growth of vortex B3 (figure 16, instant iii). Then, $W_{fi,2}$ is rising from negative to positive (process IV $^{\prime }$ ), resulting from the formation of vortex A4 on the upper side (figure 16, instants iii and iv). This is followed by the decreasing $W_{fi,2}$ from positive to negative (processes V $^{\prime }$ and VI $^{\prime }$ ), as a consequence of the growing vortices B4 and A5 (figure 16iv, v). The work done is summarized in figure 19(b). As the cylinder is on its upward movement, B2 gains energy, at least partially, from the cylinder during the entire process I $^{\prime }$ . Then, A3 gains energy first from the cylinder before the cylinder reaches $Y/D=0.3$ , but transfers energy to the cylinder afterwards (process II $^{\prime }$ ). After that, B3 delivers energy to the cylinder during process III $^{\prime }$ . This is followed by the occurrence of A4, which absorbs energy partially from the cylinder before getting the top, and the positive work is done as the cylinder goes down (process IV $^{\prime }$ ). Finally, vortex B4 transfers energy to the cylinder (process V $^{\prime }$ ) and A5 absorbs energy instead (process VI $^{\prime }$ ). Positive $W_{fi,2}$ means energy transfer from vortices to the cylinder and vice versa. The negative $W_{fi,2}$ feeds B2 (figure 16ii) and A5 (figures 16v, 15 f) with energy, resulting in their growth in size, but also accelerate the breakup of the vortices due to their intense interaction with the gap shear layers. The strong interactions between B3 (or B4) and the gap shear layers produce a positive $W_{fi,2}$ , resulting in shattered vortices B3 and B4 (figure 15 d–f). Moreover, the negative $W_{fi,2}$ during processes (III $^{\prime }$ ) and (IV $^{\prime }$ ) may be responsible for the well structured vortices A3 and A4, which do not appear to interact with the gap shear layers and decay less rapidly in the wake than the others.

The cumulative work $\sum W_{fi}\,(=\sum C_{L}\times d(Y/D))$ for fifteen cycles of the cylinder oscillation is presented in figure 18(c) to gain insight into how $W_{fi,1}$ and $W_{fi,2}$ contribute to the cylinder vibration. Evidently, after the fifteenth cycle, $\sum W_{fi,1}$ is approximately 8.1 at $t/T=15$ , a little smaller than $\sum W_{fi}\,({\approx}8.3)$ , while $\sum W_{fi,2}$ is approximately 0.2, much smaller than $\sum W_{fi}$ or $\sum W_{fi,1}$ . The observation indicates that the energy transferred from the flow to the cylinder is largely contributed from $\sum W_{fi,1}$ , rather than from $\sum W_{fi,2}$ , which is related to vortex shedding.

Finally, it is useful to check the balance between the energy dissipated by the structural damping and the energy transferred from the fluid to the cylinder. The energy $W_{f}$ transferred from the fluid to the cylinder motion for each cycle of the oscillation is given by $W_{f}=\unicode[STIX]{x03C0}F_{L0}Y_{0}\sin \unicode[STIX]{x1D719}$ , where $Y_{0}$ is the amplitude of the cylinder oscillation, $F_{L0}$ is the amplitude of the lift force, and $\unicode[STIX]{x1D719}$ is the phase shift between the lift force and cylinder displacement. The energy dissipated by the structural damping is given by $W_{d}=-\unicode[STIX]{x03C0}c\unicode[STIX]{x1D714}Y_{0}^{2}$ , where $c$ is the structural damping and $\unicode[STIX]{x1D714}=2\unicode[STIX]{x03C0}f_{osc}$ ( $f_{osc}$ is the oscillation frequency). The force associated with the fundamental frequency of the cylinder oscillation contributes most to the total force, while the components of other frequencies make a negligible net contribution to the energy transfer between the fluid and the structure (Qin et al. Reference Qin, Alam and Zhou2017). Figure 20 presents the time histories of $W_{f}$ and $W_{d}$ for the downstream cylinder at $L/D=2.5$ and $A_{d}^{\ast }=0.6$ . It is evident that $W_{f}\approx -W_{d}$ or $W_{f}+W_{d}\approx 0$ – that is, the energy transferred from fluid to structure is equal to the energy dissipated by structural damping. While $W_{d}$ closely hovers around the mean (dashed line), $W_{f}$ is almost constant, because $F_{L0}$ was not as constant as $Y_{0}$ (figure 17 b).

Figure 20.

Time histories of $W_{f}$ and $W_{d}$ for the downstream cylinder, $L/D=2.5$ , $A_{d}^{\ast }=0.6$ . The dashed line represents the time mean $W_{f}$ .