5.1 Regimes mapped across the parameter space of $U^{\ast }$ , $T/D$

We first present data maps for the entire reduced velocity–cross-stream offset parameter space, before examining particular cases. Figure 5 plots contours of maximum oscillation amplitude, $A_{MAX}^{\ast }$ , maximum coefficient of lift, $C_{L_{MAX}}$ , and mean coefficient of drag, $\overline{C_{D}}$ , over the range of reduced velocity, $0.0\leqslant U^{\ast }\leqslant 14.0$ , and cross-stream offset, $0.0\leqslant T/D\leqslant 5.0$ . The left-hand panels show data for the upstream cylinder and the right-hand panels data for the downstream cylinder. Value $T/D=0.0$ indicates the cylinder arrangement where the upstream and downstream cylinders have their origins at $\{x_{0},y_{0}\}=\{-0.75,0.0\}$ and $\{0.75,0.0\}$ ; that is, the tandem case. The maximum value of cylinder offset, $T/D=5.0$ , indicates an arrangement where the cylinders are free to oscillate in the cross-stream direction around $\{x_{0},y_{0}\}=\{-0.75,2.5\}$ and $\{0.75,-2.5\}$ . For the reduced velocity, low values of $U^{\ast }$ mean stiff elastic mountings, with $U^{\ast }=0.0$ implying entirely static non-vibrating cylinders. As reduced velocity, $U^{\ast }$ , increases, the elastic mounting becomes slacker.

Figure 5.

Over the $U^{\ast },T/D$ parameter space, (a,b) contours of maximum amplitude of oscillation, $A_{MAX}^{\ast }$ , (c,d) contours of maximum coefficient of lift, $C_{L_{MAX}}$ , and (e,f) contours of mean coefficient of drag, $\overline{C_{D}}$ . Maxima are calculated as an average of the greatest 10 % of peaks. Panels on the left are for the upstream cylinder and on the right for the downstream.

The two contour maps for maximum oscillation amplitude, $A_{MAX}^{\ast }$ , show those arrangements that result in the largest displacements for the front and rear cylinders. For the front cylinder, the largest excitation occurs at low and zero cross-stream offset, $T/D$ , and in the range $5.0\leqslant U^{\ast }\leqslant 8.0$ . When the offset is increased further, the maximum amplitude decreases before converging towards the result for a single isolated cylinder, which is evident from the increasingly vertical contour lines, with $T/D$ . For the higher reduced velocities, the maximum amplitude oscillation occurs around $T/D=0.0$ and is greater than the result of the single isolated cylinder.

As evident from the darker contours, the rear cylinder oscillates at a greater amplitude than the front cylinder for large parts of the parameter space. The greatest amplitude occurs for low values of $T/D$ and in the reduced velocity range $6.0\leqslant U^{\ast }\leqslant 11.0$ .

The contours of maximum coefficient of lift, $C_{L_{MAX}}$ , show the highest values for $U^{\ast }=3.0$ and $4.0$ , and intermediate transverse $T/D\simeq 1$ . It is proposed that this is due to the strong influence of the flow and subsequent vortex formation through the gap between the cylinders, similar to the mechanism of excitation for wake-induced vibration proposed by Assi et al. (Reference Assi, Bearman and Meneghini2010); this point is further expanded on in § 5.4.

For the mean coefficient of drag, $\overline{C_{D}}$ , the two cylinders give similar contour maps, except for those cases where the downstream cylinder lies mostly, or entirely, in the wake of the upstream cylinders, thereby hiding it from the oncoming free-stream flow. The cases producing the lowest mean drag on the rear cylinder are the tandem cases at low and zero reduced velocity; for these cases, the amplitude of oscillation is very small, meaning the rear cylinder never, or hardly, comes out from behind the upstream cylinder. Drag increases on the downstream cylinder as the offset, $T/D$ , increases, resulting in the downstream cylinder meeting more of the free-stream flow, and as the reduced velocity increases. As the reduced velocity increases, the amplitude of oscillation increases, with the increased drag on the downstream cylinder indicative of a phase difference between the two cylinder oscillations, which is discussed more later in the study (see § 5.3).

There are two broad regimes in the parameter space which can be differentiated by the orientation of the contours in figure 5. For higher values of reduced velocity, $U^{\ast }>8.0$ , that is, less spring stiffness, the contours are mostly aligned horizontally. This indicates a stronger dependence of behaviour on cylinder cross-stream offset, $T/D$ , than on reduced velocity, $U^{\ast }$ , or spring stiffness. For reduced velocity, $U^{\ast }<8.0$ , stiffer springs, the dependence is less on offset, $T/D$ , and more on $U^{\ast }$ . The exception to both these categorisations is the section of parameter space around $T/D=1.0$ . Here, there is strong variation with offset, around for example $U^{\ast }=5.0$ , and some variation with reduced velocity, around for example $U^{\ast }\approx 10.0$ .

Across the parameter space, there are a number of frequencies present in the dynamics of the system, the effects of which vary throughout. Firstly, for an isolated, single cylinder at Reynolds number $Re=200$ , the vortex shedding frequency, or Strouhal frequency, is $St=0.198$ . For rigid cylinders, with increasing cross-stream offset, the frequency of the lift force on both cylinders is expected to approach this value. When the cylinders are close to each other, the vortex formations will be affected, so the frequency will likely change. As reduced velocity increases, the natural frequency of the elastic mounting, $f_{N}(D/U)=1/U^{\ast }$ , initially becomes more important. As reduced velocity increases further, the forcing of the spring becomes weaker and the cylinders are more free to move in response to the fluid forcing. For $U^{\ast }=\infty$ , the cylinders are entirely free to move and effectively have no equilibrium position.

Figure 6.

The frequency content of the lift signal on the rear cylinder across the offset, $T$ , range and for eight values of reduced velocity, $U^{\ast }$ . Each plot is formed by calculating the frequency spectrum for each case of $T/D$ , normalising each spectrum by its maximum, and then ‘stacking’ the spectra next to each other. Dark regions represent intensity of a given frequency in the lift signal for a given $T/D$ . Contour values vary from white, $1\times 10^{-6}$ , to black, $1$ . Also shown with dots is the difference in primary frequencies of the two cylinders.

Figure 6 is a compilation of the frequency content of the lift force on the rear cylinder, across the range of $T/D$ tested and for eight values of $U^{\ast }$ . To construct the plots, the frequency spectra for each case of $T/D$ are normalised by their maximum and then stacked next to each other, thereby showing the variation of frequency intensity in the fluid forcing as the offset is changed. Looking first at the plot for reduced velocity, $U^{\ast }=0.0$ , for the tandem case, $T/D=0.0$ , there is a single dominant frequency, $f(D/U)=0.169$ . This is lower than the value for a single isolated cylinder, $0.198$ . The single frequency is due to the fact that there is no flow through the gap between the cylinders, and the two cylinders effectively behave as a single body with vortex shedding from the rear cylinder only. As offset increases, this frequency initially decreases, then increases again, until at offset of approximately 1.4 a broadband response is evident, indicating a quasi-periodic or chaotic fluid forcing and flow; the frequency content here will be investigated further in the following section (§ 5.2). As offset increases towards 5.0, the single dominant frequency emerges again, converging to the frequency for a single cylinder.

Note that the term ‘chaos’ here refers to a response that is not periodic, and has a spectrum that contains at least three incommensurate frequencies. Such a response does not appear to repeat over any time scale, and as such this definition is the same as that used in dynamic systems.

For the cases shown where the cylinders are free to vibrate, $U^{\ast }=3.0$ , 4.0, 5.0, 6.0, 8.0, 10.0 and 14.0, a range of responses are evident. The dark areas represent broadband response, indicating quasi-periodic or chaotic flows. The two maps most qualitatively similar are the maps for reduced velocities, $U^{\ast }=0.0$ and $14.0$ . Out of the cases shown, these two cases are effectively the cases where the frequency and forcing of the elastic mounting are the weakest, i.e. where there is effectively no response at the body’s natural frequency. For $U^{\ast }=0.0$ , the cylinders do not vibrate at all, while for $U^{\ast }=14.0$ , the springs are slack, meaning that, for both, the fluid forcing dominates, thereby giving similar frequency content maps (note that the data for $U^{\ast }=0.0$ have higher resolution in $T/D$ than the other reduced velocities tested).

An example case where the elastic mounting does play a significant role is reduced velocity $U^{\ast }=5.0$ . For the tandem configuration, $T/D=0.0$ , the oscillation is periodic, dominated by the primary frequency, which is slightly below the natural frequency of the structure. The other frequency that appears for zero offset is the third harmonic of the primary. Energy in these odd harmonics is indicative of the spatio-temporal symmetry in the forcing on the cylinder. Once an offset is introduced, this symmetry is broken and the second harmonic increases in strength. Periodic vortex shedding persists until the offset, $T/D$ , is equal to ${\approx}0.6$ . In the cylinder offset range $0.6\leqslant T/D\leqslant 1.8$ , the spectra of the lift coefficient signal is not periodic. The more broadband spectrum indicates irregular, perhaps chaotic, vortex shedding. Beyond $T/D\approx 1.7$ , the shedding becomes more regular, with the broadband noise decreasing. In the range $1.7\leqslant T/D\leqslant 2.4$ , the second and third harmonics are distinct as well as frequencies spaced at one quarter of the primary frequency, indicative of quasi-periodic or period-doubling behaviour. This pattern abruptly ends for $T/D=2.4$ and the vortex shedding becomes dominated by a single frequency.

Figure 7.

The parameter space, mapped by oscillation frequency content of the coefficient of lift signals of the two cylinders. Symbols are the following: solid black points denote periodic flows, with both cylinders oscillating at the same frequency; hollow black dots denote periodic oscillations, but with period-doubling, or P $_{2}$ , oscillations; hollow red dots denote periodic oscillations, with period-multiplying P $_{N>2}$ oscillations; solid red dots denote quasi-periodicity; while blue dots represent chaotic oscillations.

Figure 7 provides a summary of the temporal behaviour over the parameter space, where individual cases have been inspected and categorised as being periodic on the period of vortex shedding, periodic over some multiple of the period of vortex shedding, quasi-periodic, or chaotic. Figure 7 gives an overview of the temporal behaviour of the cylinder oscillations over the parameter space; examples of specific cases appear later in the following subsections. Solid black points represent cases where the oscillations are periodic and synchronised. Hollow black points represent cases where the oscillations are still periodic and synchronised, but are period doubling, the flow repeating over two cycles of oscillation. These oscillations are designated as P $_{2}$ . The hollow red points present oscillations that are again periodic, but repeat over $N$ cycles, where $N>2$ . The distinction between P $_{2}$ and P $_{N>2}$ oscillation has been made to represent the increasing period of the oscillations as $N$ increases, which quickly becomes difficult to differentiate from a quasi-periodic response.

The solid red points represent cases where the oscillations of the cylinders are quasi-periodic. This includes quasi-periodicity arising from disordered vortex shedding, but also from small differences in the primary frequency of the oscillation of the two cylinders. For example, the periodic synchronised oscillation for $U^{\ast }=5.0$ and $T/D>2.5$ is only present for a narrow range of reduced velocity. For the other reduced velocities tested, at $T/D=4.0$ and $5.0$ , the oscillations are strictly quasi-periodic. However, this is due to each cylinder effectively shedding independently, but with a small difference persisting in the frequency of vortex shedding of the upstream and downstream cylinders. The wakes are interacting in that the presence of one cylinder slightly changes the vortex shedding frequency of the other, but that is the extent of the interaction. In figure 7, we have not attempted to distinguish between weak and strong quasi-periodicity. The frequency content plots of figure 6 give some indication as to the level of complexity of the flow and body oscillations over the parameter space in the quasi-periodic regions.

Also plotted in figure 7 are solid blue points; these represent cases classified as chaotic. The distinction between quasi-periodic and chaotic flows is made chiefly on inspection of Lissajous plots of the cylinder displacements (plots of the displacement of the front cylinder as a function of the displacement of the rear cylinder); where the orbits do not immediately appear simply modulated, we have classified the flow as chaotic. This distinction is subjective; therefore the boundary between solid blue and red points on figure 7 is fuzzy. Nonetheless, it serves as an approximation of regions of the parameter space where the flows are the most disordered.

Figure 8.

The parameter space, mapped by oscillation and vortex shedding mode. Shaded grey areas denote regions of the parameter space where cylinder oscillations are periodic and synchronised.

Figure 8 attempts to categorise each case by the type of vortex formation on each cylinder, based on such information as which cylinder sheds or does not shed vortices, whether there is significant flow in the gap between the cylinders and which vortices pair. For the tandem cases we have adopted a similar classification as used by Borazjani & Sotiropoulos (Reference Borazjani and Sotiropoulos2009), describing the low reduced velocity $U^{\ast }<5.0$ cases as mode 1. Borazjani & Sotiropoulos (Reference Borazjani and Sotiropoulos2009) denoted $U^{\ast }\,\geqslant 7.0$ cases as state 2, and flows in the range $5.0\leqslant U^{\ast }\leqslant 6.0$ as a critical state, between states 1 and 2. Instead, we choose to denote their state 2 as mode 3, and the critical state as mode 2. Discussion of these tandem modes will be continued in § 5.3.

The tandem modes only survive a short time for small cross-stream offsets. Modes $2_{1}$ and $3_{1}$ are one-sided versions of the tandem modes. The region marked as ‘Gap flow dominated’ indicates states where the flow between the cylinders dominates, with the flow characterised by pairing of either shear layers or vortices from the inside side of each cylinder. This region is centred around offset $T/D=1.0$ , and is generally characterised by smaller-amplitude and quasi-periodic or chaotic oscillations. At greater cylinder offset, the ‘Wake pair dominated’ region occurs. Instead of gap pairs of vortices and shear layers dominating, the flow is composed of pairs of vortices shed from the same cylinder in a similar manner to the shedding from a single isolated cylinder.

Regions of periodic synchronised oscillation (both cylinders having the same primary frequency) identified in figure 7 are included as greyed regions on figure 8. The cylinders (or lift coefficient in the case of the rigid $U^{\ast }=0.0$ cases) oscillate at the same primary frequency when reduced velocity and cross-stream offset are both near zero. Periodicity occurs for high reduced velocity and low offset and tandem cases, and for a narrow band of reduced velocity at offset $T/D>2.0$ .

Interestingly, when the cylinders are in close proximity, for $T/D<2$ , the oscillation is disordered and not periodic around $U^{\ast }=5$ . It is here where the natural structural frequency of the bodies is close to the vortex shedding frequency of a single isolated cylinder. This is in stark contrast to the single isolated cylinder, which will synchronise or undergo lock-in when these two frequencies match, as do the two cylinders for larger offset. This is perhaps not surprising: when the bodies are close, they will each strongly influence the vortex formation and shedding from the other, changing the time scale and therefore the frequency; there are other time scales to consider, such as the advection time across the gap; and there are new physical phenomena at play that do not occur for the single cylinder, especially the complexity introduced by the flow between the cylinders.

To provide further detail of these identified modes and regions, the following subsections treat subregions of the parameter space. First, § 5.2 examines the limiting case of rigid cylinders, or a line along $U^{\ast }=0$ in figure 8.

5.2 The $U^{\ast }=0$ limiting case: stationary cylinders

Stationary cylinders represent a limiting case of (2.1) where the reduced velocity $U^{\ast }=0$ , resulting in no flow-induced vibration of the cylinders.

Figure 9.

For $L=1.5$ , the variation with cross-stream offset, $T/D$ , of the mean values of the drag and lift coefficients for the front and rear cylinders, denoted by subscripts 1 and 2, respectively. The horizontal dashed lines mark the average lift coefficient $\overline{C_{L}}$  and average drag coefficient $\overline{C_{D}}$  for a single isolated cylinder ( $\overline{C_{L}}=0$ and $\overline{C_{D}}=1.384$ , adjusted for equivalent blockage ratio). Vertical dashed lines represent the boundaries between mode 1, gap flow and wake pair dominated regions, as on figure 8.

Figure 9 plots the variation with cross-stream offset of the mean values of the lift and drag coefficients for both cylinders. For the tandem cylinders ( $T=0.0$ ), most of the drag is experienced by the front cylinder; the rear cylinder experiences a negative mean drag. The negative mean drag on the rear cylinder only occurs for offsets $T/D<0.2$ . The drag on the rear cylinder increases almost linearly up to offset $T/D=1.0$ , from which point it exceeds the drag on the front cylinder.

The mean lift force is zero for the tandem $T=0.0$ configuration. With increasing offset, the mean lift becomes positive for the rear cylinder and negative for the front, indicating a mean force on the cylinders away from the centreline. Around $T/D=1.2$ , the lift on the front cylinder becomes positive again.

Figure 9 only describes the mean forces on the cylinders. Figure 10 plots the variation with offset $T/D$ of the intensity across the spectrum frequency in the coefficient of lift signal, for both cylinders, replotting figure 6(a), but including the data for both the upstream (top) and downstream (bottom) cylinders. The primary frequency of the vortex shedding in each case can be identified by the darkest sections of the plot. Immediately apparent is that, for $T/D\lesssim 0.6$ (i.e. mode 1 according to figure 8), the vortex shedding frequency from both cylinders is similar, and is less than the vortex shedding (Strouhal) frequency of the single isolated cylinder, $St=0.198$ . This corresponds well to the first base ‘mode’ identified from previous literature and described in the introduction (§ 1), for $T/D\lesssim 0.4$ , where there is no gap flow between the cylinders.

Figure 10.

For $L/D=1.5$ and fixed cylinders ( $U^{\ast }=0$ ), the frequency content of the lift force as a function of $T/D$ for both cylinders. The plots are formed in the same way as described in figure 6. The horizontal dotted line indicates the Strouhal frequency for a single isolated cylinder. Vertical dashed lines represent the boundaries between mode 1, gap flow and wake pair dominated regions, as in figure 8.

For $0.6\leqslant T/D\lesssim 1.5$ , figure 10 shows that the primary frequency of the flow starts to increase with increasing $T/D$ , and a stronger component at twice the primary frequency also appears. This corresponds to the second ‘base’ mode described in § 1 identified from the literature for $0.4\leqslant T/D\leqslant 1.3$ , and characterised by a flow through the gap between the cylinders that becomes stronger with increasing $T/D$ . The presence of a frequency at twice the primary frequency is a direct consequence of the broken symmetry due to the non-zero offset – each vortex shed in the wake has an impact on the gap flow, which induces a force in the same direction regardless of the sign of the vortex, hence the doubling of the frequency.

For $T/D\gtrsim 1.5$ , with the transition to a wake pair dominated flow according to figure 8, a more complex behaviour appears. This results from the presence of a difference in primary vortex shedding frequency on each cylinder, and the close proximity of these vortex wakes occurring at different time scales. Again, this is consistent with the third ‘base’ mode described from the literature for $T/D>1.3$ in § 1. In these cases, the upstream cylinder is now forming and shedding its own vortex pairs. These vortices have a smaller formation area because of the presence of the downstream cylinder, which results in smaller vortices being shed at a higher frequency than they do from the downstream cylinder. Figure 10 shows that the flow can be highly complex, with a frequency response consisting of a series of well-defined harmonics, or broadband noise in the range $1.5<T/D<1.8$ , the focus of the inset of the spectrum plot for the rear cylinder. The difference in primary frequency decreases with increasing $T/D$ , until they both converge to the value for a single cylinder at large values of $T/D$ . The phenomenon of different Strouhal frequencies has also been noted in Alam & Sakamoto (Reference Alam and Sakamoto2005), in a study of flow past staggered cylinders. For non-tandem and non-side-by-side arrangements, they observed, for a far greater Reynolds number than the current study, $Re=5.5\times 10^{4}$ , sometimes intermittently, different Strouhal frequencies for particular cases, describing the flow as bi- or multistable.

This range of cross-stream offset $0.0\leqslant T/D\leqslant 5.0$ was also covered numerically in the work of Tong, Cheng & Zhao (Reference Tong, Cheng and Zhao2015) for $Re=10^{3}$ , but traversed in terms of angle and pitch ratio between the two cylinders (see their figure 4). The description of the regimes provided here is consistent with their figure 18 along a traverse at $L/D=1.5$ .

Figure 11.

For stationary cylinders, $U^{\ast }=0$ , plots of vorticity contours, time series of the coefficient of lift of each cylinder and the corresponding spectra, red corresponding to the upstream cylinder and green the downstream, for cylinder offsets $T/D=0.0$ , $1.0$ , $2.3$ and $3.5$ . Contours vary from blue to red in the range $-1\leqslant \unicode[STIX]{x1D714}_{z}D/U\leqslant 1$ .

Figure 11 plots details of four cases, picking out the various regimes of behaviour classified in figure 8 and further described in figures 9 and 10. The flows for offset $T/D=0.0$ correspond to mode 1 where the front cylinder does not shed vortices and only one vortex street is present. Offset $T/D=1.0$ however corresponds to the gap-dominated flow regime and the flow is more complex with stronger contributions from all harmonics of the primary frequency present in the lift coefficient signal of each cylinder. Despite the gap flow, the cylinders still largely shed vortices as one body. In both cases the vortex formation region is larger than for the single isolated cylinder, resulting in a lower vortex shedding frequency. For $T/D=2.3$ , the flow is now wake pair dominated. A periodic beating is evident in the time series of the cylinder oscillation, indicative of the difference in primary vortex shedding frequency. For the case of $T/D=3.50$ , the frequency difference persists, but is much smaller, resulting in a weaker, longer-period beating in the signal as the interaction between the wakes lessens as the cylinders are moved further apart.

Figure 12 focuses on cases where there is a significant difference between the primary frequencies, spanning the range $1.3\leqslant T/D\leqslant 1.9$ , the same range as shown in the inset in figure 10. This range encompasses the transition from gap pair dominated to wake pair dominated flow. The inset shows a frequency content strongly dependent on cross-stream offset. Data have been obtained at a high resolution of $T/D$ , and figure 12 presents six example flows in this region. Figure 12 shows the complexity of the wake and the subtle shifts that can occur with small changes in the offset, as the flow generated through the gap between the cylinders is highly sensitive.

Figure 12.

For stationary cylinders, $U^{\ast }=0$ , plots of vorticity contours, time series of the coefficient of lift of each cylinder and the corresponding spectra, red corresponding to the upstream cylinder and green the downstream, for cylinder offsets $T/D=1.40$ , $1.43$ , $1.47$ , $1.51$ , $1.54$ and $1.60$ . Contours vary from blue to red in the range $-1\leqslant \unicode[STIX]{x1D714}_{z}D/U\leqslant 1$ .

The images of the flow are all taken at a similar phase, when the lift force is at a local maximum. The images show that the large-scale features of the wake are not impacted greatly by the small change in offset. However, the associated time series of lift force, and the resulting frequency spectra, show very strong variation driven by the complex flow in the gap. For example, each second case shown (offsets, $T/D=1.43,1.51$ and $1.60$ ) are P $_{N}$ -periodic, repeating over $N$ vortex shedding cycles (where $N=2,4,5$ , respectively). This is indicated by the regular modulation of the time series of the lift force and the ‘comb’ structure of the frequency spectra, with frequencies occurring at increments that are $f/N$ , where $f$ is the vortex shedding frequency. In fact, $N$ can be discerned by counting how many spikes occur in the spectrum before the primary frequency. The shedding locks on to these P $_{N}$ oscillations for extremely narrow ranges of cross-stream offset. These periodic cases are interleaved with quasi-periodic or even chaotic responses at offsets $T/D=1.40,1.47,1.54$ . In these cases, the modulation in the time series is less regular, and the frequency spectra are more broadband, with an essentially continuous distribution of energy across frequencies. This interleaving of P $_{N}$ -periodic with quasi-periodic and chaotic responses leads to an incredibly dense variation of frequency content (see the inset of figure 10). Such a dense variation does not seem to persist for the vibrating cylinders (see figure 6), or for the qualitatively similar $U^{\ast }=14.0$ case; in these cases the cylinders are not rigidly held and hence the vortex shedding frequencies are not rigidly tuned to the cross-stream offset.

5.3 The $T/D=0$ limiting case: tandem cylinders

This section examines the case of vibrating cylinders aligned in a tandem configuration, $T/D=0$ ; from figure 8 traversing the bottom edge of the parameter space. Borazjani & Sotiropoulos (Reference Borazjani and Sotiropoulos2009) presented results for this $T=0.0$ and $L/D=1.5$ case, for the reduced velocity range $3\leqslant U^{\ast }\leqslant 14$ . That work is compared and expanded on in this section. Figure 13 plots a direct comparison of the slice of the parameter space of the current study, at $T/D=0.0$ , with the work of Borazjani & Sotiropoulos (Reference Borazjani and Sotiropoulos2009), showing the variation of maximum displacement, and of r.m.s. values of the coefficients of drag and lift for each cylinder. Also shown are comparison data of a single isolated vibrating cylinder from both the current study and Borazjani & Sotiropoulos (Reference Borazjani and Sotiropoulos2009). The $A_{MAX}^{\ast }$ value is calculated as the mean of the greatest 10 % of local peaks of $Y$ . The general picture given by the current results and those of Borazjani & Sotiropoulos (Reference Borazjani and Sotiropoulos2009) is the same across the three scalars presented in figure 13, providing a level of confidence in the results obtained. However, there are some differences that need to be addressed. For the single isolated cylinder, the values of $A_{MAX}^{\ast }$ for the range $4\leqslant U^{\ast }\leqslant 7$ , for the current study, are greater than those from Borazjani & Sotiropoulos (Reference Borazjani and Sotiropoulos2009). For the tandem arrangement, the oscillation amplitude of the front cylinder is greater, while that of the rear cylinder is reduced slightly for $5\leqslant U^{\ast }\leqslant 7$ . Similarly for the lift and drag, some variation between the present results and those of Borazjani & Sotiropoulos (Reference Borazjani and Sotiropoulos2009) can be seen.

Figure 13.

For $T/D=0.0$ and $L/D=1.5$ , the variation with $U^{\ast }$ of the maximum displacement of each cylinder, $A_{MAX}^{\ast }$ , the phase difference between the cylinder oscillations, $\unicode[STIX]{x1D6FE}$ , and the r.m.s. values of the drag and lift coefficients. Solid symbols represent results from the current study, while hollow ones are from Borazjani & Sotiropoulos (Reference Borazjani and Sotiropoulos2009).

Attempts to eliminate this variation by trying to match the numerical set-up between the two studies were made. Borazjani & Sotiropoulos (Reference Borazjani and Sotiropoulos2009) used an inlet length and lateral clearance of $8D$ and an outlet length of $24D$ . Therefore, there is a significant difference in blockage ratio between the two studies: for a single cylinder, 6.25 % against 3.33 % for the current study. The code for the current study, run on this restricted domain, produced only marginal differences in the returned results for $A_{MAX}^{\ast }$ . There is also a significant difference in time step used between the two studies: $\unicode[STIX]{x1D6FF}t=0.02$ in Borazjani & Sotiropoulos (Reference Borazjani and Sotiropoulos2009) and $\unicode[STIX]{x1D6FF}t=0.004$ in the current. This difference is mostly due to the higher grid resolution used in the current study. To examine the effect of time step, we have also run both grid domain sizes using time steps of $\unicode[STIX]{x1D6FF}t=0.004$ and $0.002$ , to examine any sensitivity to temporal resolution. We found no significant effect resulting from this change in time step. We ran further tests at lower grid resolution and larger time step; the results for $A_{MAX}^{\ast }$ returned by the $M8$ grid for a time step up to $\unicode[STIX]{x1D6FF}t=0.0125$ were greater than those returned for the higher-resolution grid. The loose coupling scheme outlined in Borazjani & Sotiropoulos (Reference Borazjani and Sotiropoulos2009) was also implemented in the current code, but had negligible effect on the results. In short, it is not completely clear why this discrepancy exists. We note that the single cylinder results from the current study match very closely with the numerical results from Leontini et al. (Reference Leontini, Thompson and Hourigan2006) (see Griffith et al. Reference Griffith, Lo Jacono, Sheridan and Leontini2016), which employed a highly accurate spectral-element method.

Figure 14 plots Lissajous curves of lift coefficient and displacement across the $U^{\ast }$ range. These curves show a strong similarity to those of figure 8 of Borazjani & Sotiropoulos (Reference Borazjani and Sotiropoulos2009). The main differences here are that the orbits are much closer to symmetric around $Y=0$ and $C_{L}=0$ in the current results (which may account for some of the differences between the two studies) and also the more meandering paths of the cases for $U^{\ast }=5$ and $6$ , indicative of some quasi-periodicity.

Figure 14.

For the tandem arrangement with zero offset $T/D=0.0$ , Lissajous curves of lift coefficient and displacement for $U^{\ast }=3$ , $4$ , $5$ , $6$ , $7$ , $8$ , $10$ and $14$ . Red represents the front cylinder and green the rear cylinder.

It is shown below that these cases, $U^{\ast }=5$ and $6$ , occur in a distinct shedding regime where vortices formed at the rear of the front cylinder are forced through the gap between the cylinders to one side of the rear cylinder at one point in the oscillation cycle, and then to the other side of the rear cylinder a half-cycle later, as the two cylinders oscillate essentially out of phase. The result of this is that the lift forces on, and the resulting motions of, the two cylinders come from complex vortex–vortex and vortex–structure interactions, similar to the classic wake-induced vibration description provided in Assi et al. (Reference Assi, Bearman and Meneghini2010). Hence, it is probably unsurprising that there is not an exact match between the two studies for these cases.

For zero cross-stream offset, $T/D=0.0$ , Borazjani & Sotiropoulos (Reference Borazjani and Sotiropoulos2009) defined two broad flow states: state 1 refers to those cases where the front cylinder oscillates with larger amplitude than the rear, for $U^{\ast }\leqslant 4.0$ ; state 2, where the rear cylinder oscillates with larger amplitude, for $U^{\ast }\geqslant \approx 7.0$ . They classed the cases in between these two ranges as ‘critical’ cases in the transition between the two states. Using the vortex shedding pattern classification of Williamson & Roshko (Reference Williamson and Roshko1988), state 1 consists of a 2S vortex shedding pattern, where a single vortex of each sign is shed from the rear cylinder, with the rear cylinder enveloped by shear layers beginning at the front cylinder. State 2 consists of a 2P vortex shedding pattern, with vortices of each sign shed from both cylinders with complex vortex interaction and merging in the near wake. The ‘critical’ case for $U^{\ast }=5.0$ shows similarity to state 2, but does not show a 2P ordering, with complex interactions resulting from fast-moving vortices merging with earlier-shed vortices, producing a less ordered wake.

As outlined in the discussion of figure 8, it is proposed that the two states described by Borazjani & Sotiropoulos (Reference Borazjani and Sotiropoulos2009) are better described as regimes or modes and that the ‘critical’ cases ( $U^{\ast }=5.0$ and $6.0$ ) are in fact a distinct mode, rather than a transition case. The distinctions can be observed in the different $Y{-}C_{L}$ orbits of figure 14 and the similarities between the orbits within the ranges of $U^{\ast }$ just described.

Plots of vorticity contours throughout the oscillation for representative cases of each of the three modes are shown in figure 15. For mode 1 cases, shown for $U^{\ast }=4.0$ , the rear cylinder sits in the wake of the front cylinder throughout the oscillation cycle. The oscillation of the rear cylinder is small compared to the front cylinder (see figure 13). Vorticity that forms on the front cylinder merges with vorticity of the same sign on the rear cylinder. Vortices are shed only from the rear cylinder. One vortex of each sign is shed in each cycle, forming a regular vortex street. Given that the vortices are shed from the rear cylinder, we denote the shedding mode as 2S $_{R}$ . The mode exists for cases of $U^{\ast }=3.0$ and $U^{\ast }=4.0$ , with simulations requiring long run times ( $750U^{\ast }$ ) to reach a statistically converged solution.

Figure 15.

For the tandem cylinder arrangement ( $T/D=0.0$ ), vorticity contours for three cases, $U^{\ast }=4.0$ , $5.0$ and $8.0$ , describing the three vortex shedding modes, 2S $_{R}$ , 2P $_{F}$ and 2P. Each series of images depicts an oscillation cycle of the primary frequency, beginning with the front cylinder at its maximum displacement. Contours vary from blue to red in the range $-1\leqslant \unicode[STIX]{x1D714}_{z}D/U\leqslant 1$ .

For $U^{\ast }=5.0$ and $6.0$ , the rear cylinder oscillates more, coming out fully from behind the front cylinder. The cylinders oscillate out of phase, $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x03C0}$ (see figure 13). In contrast to mode 1, vortices do not form on the rear cylinder; instead only a small amount of vorticity is deposited in a trail. A vortex pair forms in the (periodically) larger space now available between the cylinders. This larger space is now available due to the elastic mounting of the springs, despite the close proximity of the cylinders ( $L/D=1.5$ ). This large space allows the formation of vortices in the gap, creating a wake-induced vibration on the rear cylinder (Assi et al. Reference Assi, Bearman and Meneghini2010). As the vortex pair forms, it is cut off by the rear cylinder, its orientation rolling (see P $_{1}$ in the second to fourth images of figure 15), before meeting free-stream fluid and convecting at high velocity downstream (P $_{1}$ in the fifth to eight images of figure 15). The vortex pairs overtake earlier pairs, merge and form an irregular vortex street. Because of the two vortex pairs formed in each cycle, which mostly form on the front cylinder, we denote this shedding mode 2P $_{F}$ . Apparent in the contour fields, plotted over one period of the dominant frequency, there is some quasi-periodicity; tracking down the images from top to bottom, one can see that the topology of the vorticity field is not going to repeat in the next period. For that reason, we denote this case as the period-doubling oscillation case on figure 7.

Mode 3 exists for $U^{\ast }\geqslant 7.0$ and is characterised by large oscillation of the rear cylinder, with the oscillation of the rear cylinder lagging the front cylinder by $\unicode[STIX]{x1D6FE}\simeq \unicode[STIX]{x03C0}/2$ . Similar to mode 2, the rear cylinder traverses the wake of the front, but, instead of a vortex pair forming in the gap, the rear cylinder effectively cuts through the shear layers of the front cylinder (the third to fifth images of figure 15). In a complex interaction, negative and positive vorticity from the front cylinder is subsumed by the vortex formation on the rear cylinder each half cycle. Each cycle, this vortex mingling creates two pairs of unequal-strength vortices. To account for the complex formation of these vortex pairs, the shedding mode is denoted as a 2P. These three vortex shedding modes cover the $T=0.0$ parameter space. The next section will examine the effect of increasing a cross-stream offset, introducing the staggered vibrating cylinder arrangement.

5.4 Variation of cross-stream offset $T/D$

This section describes example cases from the $U^{\ast }$ – $T/D$ parameter space not contained along the $U^{\ast }=0.0$ and $T/D=0.0$ axes. On figure 8, the tandem modes (modes 1, 2 and 3) are marked on and near the $T/D=0.0$ axis. As the cross-stream offset increases, mode 1, featuring vortex shedding from the rear cylinder only, persists up until $T/D=0.6$ , when gap flow dominated flows appear. For modes 2 and 3, for increasing offset, these first give way to modes $2_{1}$ and $3_{1}$ , which are essentially one-sided versions of the tandem modes 2 and 3.

Figure 16 plots four cases, showing examples of the tandem modes 2 and 3 and the one-sided, or asymmetric, versions of these modes. In this figure, the period doubling of modes 2 and $2_{1}$ is evident in the presence of energy at a frequency of half the primary frequency. Mode $2_{1}$ differs from mode $2$ , in that the vortex pairs in the wake (labelled in the bottom panel of the example in figure 15) are only shed from the upstream cylinder; similarly, the rolling of the vortex pair in the gap identified for mode $2$ only occurs on one side of the cylinders. The resultant wake mode is a P $_{F}$ +S $_{R}$ , with a single vortex shed from the downstream cylinder and a pair from the front. Similarly, the shear layer cut-through that occurs for mode $3$ only occurs on one side of the cylinders for mode $3_{1}$ . This results in a periodic P+S wake for mode $3_{1}$ , rather than the periodic 2P wake for mode 3.

Figure 16.

For $U^{\ast }=5.0$ and $8.0$ , and $T/D=0.0$ and $0.4$ , plots of vorticity contours, time series of the displacement of each cylinder and the corresponding spectra for the displacements; red corresponding to the upstream cylinder and green the downstream. These four cases give examples, top to bottom, of the modes 2, $2_{1}$ , 3 and $3_{1}$ , as shown on figure 8. Contours vary from blue to red in the range $-1\leqslant \unicode[STIX]{x1D714}_{z}D/U\leqslant 1$ .

Moving further in the higher offset direction of the parameter space depicted in figure 8, a region labelled ‘Gap flow dominated’ is encountered, centred around a cross-stream offset of $T/D=1.0$ . Figure 17 presents six cases in this region, for a cross-stream offset of $T/D=1.0$ , for values of $U^{\ast }=0.0$ , 4.0, 5.0, 8.0, 10.0 and 14.0. These are cylinder configuration for which the gap flow is the predominant feature of the flow; this implies an interaction of either shear layers or vortices emanating or forming on the inside sides of the cylinders. These are shear layers and vortices formed on different cylinders, having different formation areas and exposure to the oncoming free stream. There are no pairs of alternating vortices being shed from one cylinder at a given frequency; there are several frequencies possible, tied to vortex and vorticity formation on both sides of both cylinders. Therefore, this region features both periodic flows and flows characterised by quasi-periodicity and disordered vortex shedding.

Figure 17.

For $T/D=1.0$ , plots of vorticity contours, time series of the displacement of each cylinder and the corresponding spectra for the displacements, red corresponding to the upstream cylinder and green the downstream, for reduced velocities, $U^{\ast }=0.0$ , $4.0$ , $5.0$ , $8.0$ , $10.0$ and $14.0$ . These six cases give examples of the gap flow dominated region of figure 8. Contours vary from blue to red in the range $-1\leqslant \unicode[STIX]{x1D714}_{z}D/U\leqslant 1$ .

From figure 17, the common feature of the cases shown is the gap flow. In all cases, the positive vorticity generated from the upstream cylinder completely or substantially passes through the gap between the two cylinders. Following on from this is an interaction or pairing of the shear layers and vortices on the inside sides of the two cylinders; together with the elastic mountings of the cylinders, the combination of several unrelated frequencies results in the quasi-periodic flows seen for the reduced velocity cases $U^{\ast }=4.0$ , $5.0$ and $10.0$ from figure 17. The case shown for $U^{\ast }=5.0$ exhibits a strong intermittency indicative of a chaotic flow.

A distinction needs to be drawn between these gap flow dominated cases and the tandem modes 2 and 3. The tandem modes also feature strong gap flow, but the gap is present due to the elasticity of the cylinder mountings ( $U^{\ast }$ ) – and can therefore change sides – rather than the initial position of the cylinders ( $T/D$ ). Therefore, the ‘Gap flow dominated’ region can extend over the entire range of $U^{\ast }$ .

Increasing the cross-stream offset, the region marked ‘Wake pair dominated’ on figure 8 covers the parameter space entirely for $T/D>2.0$ . In this region, instead of being determined by the interaction of vorticity in the gap, the flow is dominated by pairing of vortices shed from one cylinder. Figure 18 presents flows for the same reduced velocities as in figure 17, but for a cross-stream offset of $T/D=3.0$ . All of the flows are in the ‘Wake pair dominated’ region. Although mixing of vortex streets can occur in the far wake (see the vorticity snapshots for $U^{\ast }=4.0$ and $5.0$ ), the cylinder oscillations are determined by the pairing and interaction of the vortices formed on each cylinder. This distinguishes the categorisation from the T-I and T-II regimes described by Hu & Zhou (Reference Hu and Zhou2008) and used by Tong et al. (Reference Tong, Cheng and Zhao2015), which delineate the flow according to vortex interactions at six cylinder diameters downstream. The ‘Wake flow dominated’ categorisation refers to the forcing on the cylinder and hence on the near wake. In the cases shown in figure 18, the vorticity topology in the near wake is clearly defined by the wake pairs shed from each single cylinder.

Figure 18.

For $T/D=3.0$ , plots of vorticity contours, time series of the displacement of each cylinder and the corresponding spectra for the displacements, red corresponding to the upstream cylinder and green the downstream, for reduced velocities, $U^{\ast }=0.0$ , $4.0$ , $5.0$ , $8.0$ , $10.0$ and $14.0$ . These six cases give examples of the wake pair dominated region of figure 8. Contours vary from blue to red in the range $-\!1\!\leqslant \unicode[STIX]{x1D714}_{z}D/U\leqslant 1$ .

For all six cases, the vortices appear organised. However, only the cases for reduced velocity $U^{\ast }=4.0$ and $5.0$ are classified as periodic, as shown on figure 7. For these two cases, the cylinders oscillate at the same primary frequency, with significant amplitude. For reduced velocities $U^{\ast }=0.0$ , $8.0$ , $10.0$ and $14.0$ , the cylinders either strictly do not, or only barely, vibrate, maintaining transverse separation between the cylinders. In each case, the cylinder oscillations are not synchronised, with small differences between the primary frequencies of the two cylinders resulting from the weak, but still present and noticeable, effect of the cylinders on each other; these cases are therefore classed as quasi-periodic. The small difference in frequency can be discerned in the time series of the lift coefficient shown for the $U^{\ast }=0.0$ case, producing a corresponding long period (approximately $50$ time units) beating. At this point the separation between the cylinders is such that they are approaching the state of a single isolated cylinder, which for these reduced velocities results in quasi-periodic oscillation, due to the non-matching Strouhal frequency and structural frequency.

The distribution across the parameter space of the cases featuring periodic oscillations with the same primary frequency is interesting. For the cases where the structural frequency is close to the Strouhal frequency (reduced velocity $U^{\ast }=5.0$ ), single-harmonic periodic oscillation occurs only for high cross-stream offset, as it does for the single isolated cylinder. In contrast, for the tandem cases, the flow is characterised by period multiplying and quasi-periodicity for the reduced velocity range $4.0\leqslant U^{\ast }\leqslant 6.0$ . It is only outside of this range (which gives quasi-periodicity for the single cylinder vibration) that periodic flow occurs for the tandem, or almost tandem, configuration.