4.1.1 Synchronisation modes

The influence of the plane wall on the synchronisation mode is investigated initially by conducting simulations over two different $G/D$ (0.5 and 1) and a range of $f_{d}/f_{St^{\ast }}$ values at $A/D=0.1$ under the fixed-wall conditions. The cylinder is placed $28D$ from the inlet, which is identical to the stationary cylinder condition reported earlier. The identified synchronisation modes with $A/D=0.1$ are mapped out in figures 5(c) and 5(e). The synchronisation modes at $/D=\infty$, reproduced from Tang et al. (Reference Tang, Cheng, Tong, Lu and Zhao2017), are included in figure 5(a) for the purpose of comparison.

To better illustrate the influence of the plane-wall boundary layer on the flow, additional simulations are carried out under the moving-wall conditions, where the plane wall is forced to move at the same velocity as the steady approaching flow. The synchronisation modes are shown in figures 5(b) and 5(d). The corresponding profiles of mean horizontal velocity ($u_{mean}/u_{0}$), sampled at $x=0$ with the cylinder placed at $(x,y)=(0,0)$, are shown in figure 4.

The $f_{St^{\ast }}$ values used for normalisation in figure 5 are the frequencies of vortex shedding around a stationary cylinder at corresponding gap ratios and $Re$, which are $f_{St^{\ast }}=0.192$ at $G/D=\infty$, 0.205 at $G/D=1.0$ and 0.198 at $G/D=0.5$ under the moving-wall conditions and $f_{St^{\ast }}=0.184$ at $G/D=1.0$ under the fixed-wall conditions. Since vortex shedding is completely suppressed at $G/D=0.5$ under the fixed-wall conditions, $f_{St^{\ast }}=0.155$, obtained from $G/D=0.6$, is used to normalise $f_{d}$ at $G/D=0.5$.

Similar to the wake of an isolated oscillating cylinder, a variety of synchronisation modes are revealed as the wall is introduced. There are two major differences in the synchronisation modes. Firstly, the synchronisation modes with an odd number as the denominator ($q$) of the near-wall cylinder increase significantly, as quantified from the percentage of occurrence, which is determined by the range of synchronisation $(f_{d}/f_{St^{\ast }})$ over $f_{d}/f_{St^{\ast }}\in [1,4]$. This observation is quite different from that of an isolated cylinder where the modes with even $q$ values are the preferred modes. For example, the percentage of occurrence of the $1/2$ mode decreases monotonically with the reduction of $G/D$ from $\infty$ to 1.0 under the fixed-wall conditions, and from 1.0 to 0.5 under the moving-wall conditions, whereas the percentage of occurrence of the $1/3$ and $2/5$ modes increases noticeably with the reduction of $G/D$ from $\infty$ to 1 under the fixed-wall conditions. The flow asymmetry around the cylinder, induced by the presence of the walls (stationary and moving), is identified as the physical mechanism for the phenomenon observed above. Consistently, modes with an odd-number denominator ($q$) are featured by inclined wakes relative to the flow direction. This reasoning agrees with the interpretation by Tang et al. (Reference Tang, Cheng, Tong, Lu and Zhao2017) that the synchronisation modes for an isolated cylinder prefer an even number to an odd number as the denominator ($q$), because modes with an even-number denominator exhibits spatio-temporal symmetry similar to a Kármán vortex street.

Secondly, synchronisation modes with reducible ratios of $p/q$ (i.e. $p$ and $q$ have a common factor above 1, such as $2/4$ modes) are observed at $G/D=1.0$ under the fixed-wall condition. Those modes with reducible ratios of $p/q$ have not been reported in the isolated-cylinder case (e.g. Leontini et al. Reference Leontini, Lo Jacono and Thompson2011, Reference Leontini, Lo Jacono and Thompson2013; Tang et al. Reference Tang, Cheng, Tong, Lu and Zhao2017). The $2/4$ mode is primarily observed within the parameter space occupied by the primary synchronisation mode of $1/2$. Further investigations on the reducible mode ratios are detailed in § 4.1.4.

The synchronisation modes observed with $G/D=0.5$ under the fixed-wall condition are quite different from other cases, possibly because the suppression of vortex shedding from its stationary cylinder counterpart quenches the occurrence of the modes other than modes $0/1$, $1/1$ and $1/2$.

Figure 5.

Locations of the synchronisation modes at $A/D=0.1$ for: (a) $G/D=\infty$ (Tang et al. Reference Tang, Cheng, Tong, Lu and Zhao2017); (b) $G/D=1.0$, moving-wall conditions; (c) $G/D=1.0$, fixed-wall conditions; (d) $G/D=0.5$, moving-wall conditions; and (e) $G/D=0.5$, fixed-wall conditions. Here $f_{St^{\ast }}$ is the intrinsic vortex-shedding frequency under each condition. The synchronisation modes for $q\leqslant 10$ are listed in the legend with increasing value of the mode ratio, while OMs are represented by $+$. All cases deemed QP are shown as $\boldsymbol{\cdot }$ in the map. The number above modes $1/2$, $2/5$ and $1/3$ is the calculated occurrence percentage from the range of synchronisation $(f_{d}/f_{St^{\ast }})$ over $f_{d}/f_{St^{\ast }}\in [1,4]$.

Figure 6.

Instantaneous vorticity flow field of selected synchronisation modes for an isolated cylinder (a–f) and a near-wall cylinder under moving-wall (g–l) and fixed-wall conditions (m–r) at $G/D=1.0$. Lissajous phase diagrams of the cylinder displacement versus lift coefficient are shown next to each flow field. From top to bottom: (a,g,m) mode $2/3$, (b,h,n) mode $1/2$, (c,i,o) mode $2/5$, (d, j,p) mode $3/8$, (e,k,q) mode $1/3$, and (f,l,r) mode $1/4$.

4.1.2 Flow characteristics

The instantaneous flow fields of selected synchronised cases are compared in figure 6 through vorticity contours for $p/q=2/3$, $1/2$, $2/5$, $3/8$ and $1/3$ under three different conditions, i.e. $G/D=\infty$, $G/D=1.0$ under moving-wall conditions, and $G/D=1.0$ under fixed-wall conditions.

The wake structures for the near-wall cylinder under the moving-wall conditions are similar to those for an isolated cylinder, especially for modes $1/2$, $2/5$, $3/8$ and $1/4$. A subtle difference between them is that the wakes for $G/D=1.0$ under the moving-wall conditions incline slightly upwards from the wall in the near wake and become approximately parallel to the wall in the far wake, whereas the wakes for the isolated cylinder remain parallel to the free stream and are distributed almost evenly on either side of $y=0$. The wake inclinations under the moving-wall conditions are attributable to the positive vertical velocity gradient induced by the strong gap flow, as shown in figure 4. The average velocity directly under the gap is actually larger than the approaching velocity, so that the local flow tends to decelerate ($\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}x<0$) as it leaves the gap in the downstream direction, producing an upward velocity component ($\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}y>0$) in the near-wall region. The positive vertical velocity gradient will then convect the vortices in the wake upwards.

The widths of the wake for modes with $p>1$ appear to be larger than for the modes with $p=1$ under both isolated and moving-wall conditions. This observation can be explained by considering the interactions of vortices that are generated within $p$ pair(s) of vortex shedding in $q$ cycles of cylinder oscillation (those within the solid- and dashed-line boxes in figure 6). Only two vortices of opposite signs exist in each box for modes with $p=1$, whereas multiple pairs of vortices are present and convected downstream as a group for the cases with $p>1$. According to classical vortex dynamics (Saffman Reference Saffman1992), two vortices of opposite signs translate as a unit and two vortices with the same sign rotate around each other. The rotation of vortices with the same sign in the groups of multiple vortex pairs is the direct cause of the wider width of the wake for modes with $p>1$.

On the other hand, the most noticeable difference between the wakes obtained under the moving-wall and fixed-wall conditions is that the wakes of the cylinder under the fixed-wall conditions mainly consist of negatively signed vortices that are shed from the top surface of the cylinder, for the same reason as discussed with regard to the results shown in figure 2. In addition, the vortices (with the same sign) in the wakes of modes with $p>1$ tend to rotate around each other and merge as they are convected downstream, whereas the vortex streets (of opposite signs) in the wakes of modes with $p=1$ are almost parallel with the wall without obvious relative rotation or merging of vortices.

Figure 7.

Comparison of the instantaneous vorticity flow fields under (a) moving-wall and (b) symmetry-wall conditions at $(f_{d}/f_{St},A/D,G/D)=(2,0.1,1)$, mode $1/2$.

Although the influence of the wall boundary layer on the flow is largely removed by replacing the stationary wall with a moving wall, weak vortices can still be observed on the wall near the cylinder due to the mismatch between the flow velocity in the gap and that of the moving wall. To fully remove the weak vortices on the plane wall, additional simulations are conducted by replacing the moving-wall conditions with symmetry-wall conditions ($\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y=0$, $v=0$). The instantaneous vorticity flow fields at $(f_{d}/f_{St},A/D,G/D)=(2,0.1,1)$ under the moving-wall and symmetry-wall conditions are compared in figure 7. Overall, the wake structures behind the cylinder are very similar under those two boundary conditions, suggesting the negligible effect of the weak vortices on the flow fields behind the cylinder.

4.1.3 Force characteristics

The influence of the plane wall on the drag and lift coefficients ($C_{D}$ and $C_{L}$) is investigated for the cases with $A/D=0.1$, as an example. Figure 8 shows the variations of $C_{D,mean}$ with $f_{d}/f_{St^{\ast }}$ at $A/D=0.1$ for the isolated cylinder and near-wall cylinder ($G/D=1.0$) under the fixed-wall and moving-wall conditions. Two main features are observed: (i) the $C_{D,mean}$ values under the moving-wall conditions and the fixed-wall conditions are consistently larger and smaller, respectively, than those of an isolated cylinder over the range of $f_{d}/f_{St^{\ast }}$ investigated here; and (ii) the $C_{D,mean}$ values for synchronised flows are generally larger than those of the desynchronised flows in the vicinity of the synchronised flows.

The feature (i) is primarily due to the change of the velocities of the approaching flow local to the cylinder, induced by the presence of two different types of boundary conditions as shown in figure 4. Because $C_{D,mean}$ is normalised by the free-stream velocity $u_{0}$, any change in the velocities of the approaching flow local to the cylinder will be directly reflected by the change in $C_{D,mean}$ value. Although only minor differences in the average velocity profiles exist on the top side of the cylinder, a significant increase of flow velocity on the bottom side under the moving-wall conditions leads to an increase of local approaching flow velocity near the cylinder and thus an increase of $C_{D,mean}$. Since the gap velocity shows a significant reduction under the fixed-wall conditions, it leads to a reduction of $C_{D,mean}$ through the mechanism explained above. If the mean streamwise velocity over $y=-1.5D{-}1.5D$ at $x=0$ is quantified as a representative value for the approaching flow velocity local to the cylinder, $u_{mean}/u_{0}=0.805$, 0.647 and 0.759 are obtained under the moving-wall, fixed-wall and isolated-cylinder conditions, respectively. This result provides a quantitative support to the explanation offered above.

The observed feature (ii) is interpreted by using mode $1/2$ as an example, where the most pronounced increase in $C_{D,mean}$ is observed. This notable increase in $C_{D,mean}$ in the region of synchronisation is due to the coalescence of small vortices and, thus, a more organised wake flow, which causes an enhanced shear layer and a stronger entrainment wake (Wu et al. Reference Wu, Lu, Denny, Fan and Wu1998; Tang et al. Reference Tang, Cheng, Tong, Lu and Zhao2017). Slight increases in $C_{D,mean}$ can also be observed for other synchronisation modes, but they are not as strong as the increase in $C_{D,mean}$ for mode $1/2$.

Figure 8.

Mean drag coefficient ($C_{D,mean}$) with respect to $f_{d}/f_{St^{\ast }}$ at a fixed $A/D=0.1$ for an isolated cylinder (middle line) and near-wall cylinder with $G/D=1.0$ under moving-wall conditions (top line) and fixed-wall conditions (bottom line). The same symbols as in figure 5 are also employed here to highlight the influence of synchronisation.

Figure 9.

Variations in the mean and r.m.s. of $C_{L}$ against $f_{d}/f_{St^{\ast }}$ at $(A/D,G/D)=(0.1,1.0)$ under different wall conditions: (a) $C_{L,mean}$, moving-wall conditions; (b) $C_{L,rms}$, moving-wall conditions; (c) $C_{L,mean}$, fixed-wall conditions; and (d) $C_{L,rms}$, fixed-wall conditions. The dashed lines are from the stationary cylinder. The same symbols as in figure 5 are employed.

Figure 9 shows the variations of the mean $C_{L,mean}$ and the root-mean-square (r.m.s.) $C_{L,rms}$ with $f_{d}/f_{St^{\ast }}$ under both moving-wall and fixed-wall conditions at $(A/D,G/D)=(0.1,1.0)$. The large increase of $C_{L,rms}$ in the synchronised modes is induced by the strengthened vortices in the wake at resonance. Each synchronisation mode contains an initial stage (in which $C_{L,rms}$ continuously rises), a peak and a desynchronisation stage (where $C_{L,rms}$ significantly decreases). The width of the synchronisation range ($f_{d}/f_{St^{\ast }}$) is correlated with the magnitude of $C_{L,rms}$ under both fixed-wall and moving-wall conditions. The minimum value of $|C_{L,mean}|$ in mode $1/2$ corresponds to the maximum value of $C_{L,rms}$. This behaviour occurs because the strength of vortices that are shed from both sides of the cylinder significantly increases and becomes more uniform at synchronisations. Another interesting phenomenon is the dramatic changes in $C_{L,mean}$ and $C_{L,rms}$ in figures 9(a) and 9(b) for mode $1/3$ under the moving-wall conditions, as detailed in the zoom-in views in figure 10. According to the flow field shown in figure 10, the wakes are tilting away from the plane wall for $2.824<f_{d}/f_{St^{\ast }}\leqslant 2.827$ but are more parallel to the plane wall for $2.828<f_{d}/f_{St^{\ast }}\leqslant 2.834$. Similar features also exist for an isolated cylinder for mode $1/3$ and other modes, with $q$ being an odd number (Tang et al. Reference Tang, Cheng, Tong, Lu and Zhao2017).

Figure 10.

Instantaneous flow field for an oscillating cylinder near a moving wall at (a) $(f_{d}/f_{St^{\ast }},A/D,G/D)=(2.827,0.1,1.0)$ and (b) $(f_{d}/f_{St^{\ast }},A/D,G/D)=(2.828,0.1,1.0)$, mode $1/3$.

4.1.4 Reducible mode ratios

The influence of the plane wall on the synchronisation mode with reducible ratios is investigated by conducting simulations over a range of $G/D$ and $f_{d}/f_{St^{\ast }}$ values at a fixed $A/D=0.2$ under the fixed-wall conditions. The identified synchronisation modes with $A/D=0.2$ are mapped out in figure 11. The corresponding profiles of mean horizontal velocity ($u_{mean}/u_{0}$), sampled at $x=0$ with the cylinder placed at $(x,y)=(0,0)$, are shown in figure 4. The $f_{St^{\ast }}$ values used for normalisation in figure 11 are 0.192, 0.205, 0.197, 0.189, 0.184 and 0.155 at $G/D=\infty$, 2.0, 1.5, 1.2, 1.0 and 0.5, respectively, under the fixed-wall conditions.

Figure 11.

Locations of the synchronisation modes at $A/D=0.2$ with varying $f_{d}/f_{St^{\ast }}$ and $G/D$ under fixed-wall conditions (results at $G/D=\infty$ are from Tang et al. (Reference Tang, Cheng, Tong, Lu and Zhao2017) for purpose of comparison). Refer to figure 5 caption for more details.

Figure 12.

The spectrum of $C_{L}$, the transient trace of $C_{L}(\unicode[STIX]{x1D70F})$ (red line) with cylinder displacement $X(\unicode[STIX]{x1D70F})$ (black line), and the Lissajous phase diagram of $X(\unicode[STIX]{x1D70F})$ and $C_{L}(\unicode[STIX]{x1D70F})$ for a near-wall cylinder with $G/D=1.0$ under the fixed-wall conditions: (a) mode $1/2$, $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.292)$; (b) mode $2/4$, $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.302)$; (c) QP, $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.323)$; (d) mode $3/5$, $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.365)$; (e) QP, $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.386)$; (f) mode $4/8$, $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.417)$; (g) mode $2/4$, $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.438)$; and (h) mode $1/2$, $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.469)$.

Figure 13.

Poincaré maps for $C_{L}$ for a near-wall cylinder with $G/D=1.0$ under the fixed-wall conditions (top left to bottom right): mode $1/2$, $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.292)$; mode $2/4$, $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.302)$; QP, $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.323)$; mode $3/5$, $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.365)$; QP, $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.386)$; mode $4/8$, $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.417)$; mode $2/4$, $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.438)$; and mode $1/2$, $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.469)$.

The synchronisation modes found at $G/D=2.0$ are similar to those found in its isolated-cylinder counterpart, suggesting that the influence of the plane wall is weak at $G/D=2.0$. As $G/D$ reduces, the synchronisation modes with an odd number as the denominator ($q$) increases in the occurrence percentage, which is consistent with the results obtained with $A/D=0.1$. Most importantly, synchronisation modes with reducible ratios of $p/q$, such as $2/4$, $4/8$ and $4/6$ modes, are observed at $0.5<G/D<2$. The modes with reducible ratios of $p/q$ have not been reported for an isolated cylinder with similar parameter ranges (e.g. Leontini et al. Reference Leontini, Lo Jacono and Thompson2011, Reference Leontini, Lo Jacono and Thompson2013; Tang et al. Reference Tang, Cheng, Tong, Lu and Zhao2017). Similar to $A/D=0.1$, the $2/4$ mode is primarily observed within the parameter space occupied by the primary synchronisation mode of $1/2$. Interestingly, a transition from the primary synchronisation mode of $1/2$ to QP is observed between $f_{d}/f_{St^{\ast }}=1.292$ and 1.459 at $G/D=1.0$. The transition follows a sequence of 1/2–2/4–QP–3/5–QP–4/8–2/4–1/2 modes. Only synchronisation modes of $0/1$, $1/1$ and $1/2$ are observed with $G/D=0.5$ because of the suppression of vortex shedding from its stationary counterpart.

To further investigate the nature of the synchronisation modes with reducible $p/q$ ratios and the transition from synchronisation modes to QP, the spectrum of $C_{L}$, the transient trace of $C_{L}(\unicode[STIX]{x1D70F})$ with cylinder displacement $X(\unicode[STIX]{x1D70F})$, and the Lissajous phase diagram of $X(\unicode[STIX]{x1D70F})$ and $C_{L}(\unicode[STIX]{x1D70F})$ for the transition sequence of 1/2–2/4–QP–3/5–QP–4/8–2/4—1/2 modes at $G/D=1.0$ are examined in figure 12 with $f_{d}/f_{St^{\ast }}$ ranging from 1.292 to 1.469 and $A/D=0.2$. The primary synchronisation mode of $1/2$ is observed at $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.292)$, as shown in figure 12(a), where the dominant frequency of lift is $1/2$ and all other frequencies are integer multiples of frequency $1/2$ in the spectrum. The transient trace of $C_{L}(\unicode[STIX]{x1D70F})$ completes one cycle of oscillation in exactly two cycles of cylinder oscillation, leading to a closed loop of the Lissajous phase diagram. As $f_{d}/f_{St^{\ast }}$ is increased to 1.302, as shown in figure 12(b), the synchronisation mode of $2/4$ occurs, where $C_{L}(\unicode[STIX]{x1D70F})$ completes two cycles of oscillation in exactly four cycles of cylinder oscillation. Although the dominant frequency of $C_{L}(\unicode[STIX]{x1D70F})$ is still $1/2$, a lower-frequency component of $1/4$ appears, and all other frequencies are integer multiples of frequency $1/4$. The Lissajous phase diagram is characterised by two non-overlapping closed loops. The above features of flow response suggest that mode $2/4$ is a period doubling of mode $1/2$.

As $f_{d}/f_{St^{\ast }}$ is further increased to 1.323, the flow response (through $C_{L}(\unicode[STIX]{x1D70F})$) becomes quasi-periodic (figure 12c), which is characterised by desynchronisation of $C_{L}(\unicode[STIX]{x1D70F})$ with $X(\unicode[STIX]{x1D70F})$ and non-repeatable loops of the Lissajous phase diagram. Although the dominant frequency in the spectrum of $C_{L}(\unicode[STIX]{x1D70F})$ is still $1/2$, it is modulated by multiple irregular frequency components. A synchronisation mode of $3/5$ occurs surprisingly as $f_{d}/f_{St^{\ast }}$ is increased to 1.365, where $C_{L}(\unicode[STIX]{x1D70F})$ completes three cycles of oscillation in exactly five cycles of cylinder oscillation (figure 12d). The flow response becomes quasi-periodic again as $f_{d}/f_{St^{\ast }}$ is increased to 1.386 (figure 12e). A synchronisation mode of $4/8$ appears at $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.417)$, where $C_{L}(\unicode[STIX]{x1D70F})$ completes four cycles of oscillation in exactly eight cycles of cylinder oscillation (figure 12f). The mode of $4/8$ is judged to be a period doubling of mode $2/4$, based on the characteristics of the flow response. The synchronisation mode changes to $2/4$ and finally $1/2$ again as $f_{d}/f_{St^{\ast }}$ is further increased from 1.417 to 1.469 (figures 12g and 12h).

Poincaré maps of $C_{L}$ are used to show the transition between synchronised and QP states. For a complete cycle of cylinder oscillation, the corresponding value of $C_{L}$ was recorded. Thus $C_{L}(n)$ is plotted against $C_{L}(n-1)$ in figure 13 and $C_{L}$ was recorded over 240 cycles of cylinder oscillation. For a synchronised case, the map shows a fixed number ($=q$) of points. For example, at $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.292)$ (mode $1/2$), two points are shown in the map, while at $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.302)$ (mode $2/4$), four points are recorded. On the other hand, in the QP cases, e.g. $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.323)$ and $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.386)$ as shown in figure 13, the points appear randomly distributed over a region and the number of points increases with the recorded time duration, which indicates QP behaviour. Together with the spectrum of $C_{L}$ and the Lissajous phase diagram, it is clear that the flow is undergoing a transition from synchronised to QP states.

The response of instantaneous flow fields to the cylinder oscillation is consistent with the response observed through characteristics of $C_{L}(\unicode[STIX]{x1D70F})$. As an example, figure 14 shows snapshots of vorticity contours of a mode $2/4$ case at $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.459)$ over four cycles of cylinder oscillation. The instantaneous flow fields at $\unicode[STIX]{x1D70F}=nT$ and $\unicode[STIX]{x1D70F}=nT+2T$ would have matched exactly if the synchronisation mode in this case were a $1/2$ synchronisation. While the shear layers developed on the surface of the cylinder at $\unicode[STIX]{x1D70F}=nT$ and $\unicode[STIX]{x1D70F}=nT+2T$ are similar, the interaction patterns $(2.5<x/D<7)$ of the vortices shed from the cylinder with the shear layer formed on the wall show an obvious difference at the corresponding instants. It takes two extra cycles of cylinder oscillation for the flow fields to match at $\unicode[STIX]{x1D70F}=nT$ and $\unicode[STIX]{x1D70F}=nT+4T$. Overall, two pairs of vortices are shed from the cylinder in four cycles of cylinder oscillation, leading to a $2/4$ synchronisation in this case. The interaction of the negative shear layer developed on the wall with the positive shear layer developed on the surface of cylinder is evident in figure 14.

Figure 14.

Instantaneous flow field of a mode $2/4$ at $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.459)$ under fixed-wall conditions over four cycles of cylinder oscillation.

The introduction of a wall significantly alters the flow response around the cylinder. Possible influences of the wall on the flow response include (i) the asymmetry of the flow around the top and bottom sides of the cylinder (the mode competition) and (ii) the interaction of the shear layer generated above the surface of the wall with the vortices shed from the cylinder. The blockage effect induced by the presence of the wall leads to the formation of asymmetric shear layers on the cylinder, which subsequently affects the vortex-shedding process from the cylinder. While the interaction of the shear layers formed on the wall with vortices shed from the cylinder is observed in figure 14, the blockage effect of the wall is quantified in figure 15(a) by examining the variation of mean $u$ velocity component with $y$ sampled at $x=0$ for the same case as shown in figure 14. The mean $u$ velocity component in the gap is noticeably smaller than that on the top side of the cylinder, leading to an asymmetrical approaching flow to the cylinder. We would like to emphasise that the shear layer formed on the wall would also contribute to the flow blockage effect quantified in figure 15(a). The flow asymmetry and the shear layer formed on the wall, nevertheless, are identified as the main causes for the transition to QP through cascade of period doublings.

Figure 15.

The $u$-velocity profiles along the vertical line passing through the cylinder centre $(x/D=0)$ at $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.459)$ under (a) fixed-wall condition and (b) symmetry-wall condition.

To separate the influence of the shear layers generated above the surface of the plane wall and the influence of flow blockage effect (to first-order approximation), additional simulations are carried out at $A/D=0.2$ and 0.3, $G/D=1.0$ by imposing a symmetry-wall boundary condition on the plane wall and maintaining other conditions unchanged. The influence of the shear layers generated above the surface of the plane wall can be largely removed with the symmetry-wall condition on the plane wall.

The synchronisation modes mapped out with the symmetry-wall condition at $A/D=0.2$ and 0.3, shown in figure 16, are clearly different from those obtained under the no-slip boundary condition shown in figure 11. The major differences include: (i) the transition to QP through period doubling of mode $1/2$ disappears; (ii) the period doublings of synchronisation modes still exist both inside (mode $1/2$) and outside the parameter space otherwise occupied by the primary synchronisation mode of $1/2$; (iii) the odds of period doubling of the primary synchronisation mode ($1/2$) is increased substantially; (iv) the area covered by the primary synchronisation mode of $1/2$ increases substantially and shifts towards to the direction of high $f_{d}/f_{St^{\ast }}$; and (v) the synchronisation modes other than $1/2$, $2/4$, $2/3$ and $4/6$ largely disappear over the range of $f_{d}/f_{St^{\ast }}$ covered in this study. The regime map shown in figure 16 shares certain similarities to that of an isolated cylinder (Tang et al. Reference Tang, Cheng, Tong, Lu and Zhao2017), with a major exception that period doublings of the mode $1/2$ were not observed in Tang et al. (Reference Tang, Cheng, Tong, Lu and Zhao2017). The instantaneous flow fields illustrated in figure 17 for the case at $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.459)$ with the symmetry-wall condition on the plane wall show that no shear layers are formed on the plane wall. However, the asymmetry of the shear layers formed on the cylinder still exists, as shown by the variation of mean $u$ velocity component with $y$ sampled at $x=0$ in figure 15(b), albeit the degree of asymmetry is less severe than that observed under the no-slip boundary condition. The average velocity through the gap is slightly larger than that over the top side of the cylinder. The observations suggest that (i) the shear layer formed above the plane wall is responsible for the transition to QP at this $A/D$ and (ii) the asymmetry of the mean approaching flow around the cylinder is responsible for the period-doubling behaviours both inside and outside the parameter space otherwise occupied by the primary synchronisation mode of $1/2$, based on the comparison of the regime maps with that reported by Tang et al. (Reference Tang, Cheng, Tong, Lu and Zhao2017). To further investigate the influence of the shear layers formed on the wall, simulations are also conducted at $A/D=0.2$ and 0.3 under the moving-wall conditions. The flow regime map obtained under the moving-wall conditions (not shown here) is very similar to that shown in figure 16. Although shear layers do form in the vicinity of the cylinder under the moving-wall conditions, the strength and extent of the shear layers are substantially smaller than those observed under the fixed-wall conditions.

Figure 16.

The bifurcation diagram plotted with varying normalised excitation amplitude ($A/D$) and normalised excitation frequency $(f_{d}/f_{St^{\ast }})$ at $G/D=1.0$ under symmetry-wall conditions.

Figure 17.

Instantaneous flow field of a mode $2/4$ at $(A/D,f_{d}/f_{St^{\ast }})=(0.2,1.459)$ under symmetry-wall conditions over four cycles of cylinder oscillation.

Figure 18.

The bifurcation diagram plotted with varying shear parameter ($k$) and normalised excitation frequency ($f_{d}/f_{St^{\ast }}$) at $A/D=0.2$ for an isolated cylinder in uniform linear shear flow and a near-wall cylinder at different $G/D$ under fixed-wall conditions: (a) linear shear flow, $k=0.2$; (b) $G/D=1.0$, equivalent $k=0.13$ (data extracted from figure 11); (c) linear shear flow, $k=0.1$; (d) linear shear flow, $k=0.05$; and (e) $G/D=2.0$, equivalent $k=0.02$ (data extracted from figure 11).

The results presented so far (see figures 11 and 16) show that the flow asymmetry and the interaction of the plane-wall boundary layer with vortices shed from the cylinder are two potential culprits for transition to QP through a cascade of period doublings under the fixed-wall conditions. We are, however, still unclear how the level of flow asymmetry affects the flow synchronisation modes. To quantify the influence of the level of flow asymmetry, further simulations are carried out for steady linear shear approaching flow around an isolated cylinder undergoing streamwise oscillations. The shear flow in the simulations is generated by imposing $u(y)=u_{c}+Ky$, $v=0$ on the inlet boundary, where $u_{c}$ is the approaching velocity at $y=0$ (with $x$-axis passing through the centre of the cylinder) and $K$ is the transverse velocity gradient. Along the outlet boundary, $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y=K$ and $\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}x=0$ are enforced. A no-slip boundary condition (i.e. $u=0,v=0$) is specified on the cylinder surface. Boundary conditions of $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y=K$ and $v=0$ are applied on the top and bottom boundaries, which are located $4D$ either side of the centre of the cylinder, following that used by Lei, Cheng & Kavanagh (Reference Lei, Cheng and Kavanagh2000a). The relatively small width of the computational domain is used in order to generate large shear levels in the flow and avoid reverse flow in the computational domain. The shear parameter is defined as $k=KD/u_{c}$ and $Re$ is defined using $u_{c}$. The numerical test is conducted over $(f_{d}/f_{st\ast },k)=(1.2{-}2.0,0.05{-}0.2)$ at $Re=175$ and $A/D=0.2$.

Figure 18 shows the bifurcation diagram plotted with varying $k$ and $f_{d}/f_{St^{\ast }}$ at $A/D=0.2$ for an isolated cylinder in a uniform linear shear flow and a near-wall cylinder under the fixed-wall condition. An equivalent $k$ value for the near-wall cylinder is determined by

(4.1)$$\begin{eqnarray}\displaystyle k=\frac{1}{u_{c}}\int _{G}^{G+D}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}y}\,\text{d}y, & & \displaystyle\end{eqnarray}$$

with the velocity gradient obtained from the velocity profile given in figure 3, where the cylinder is removed and $y=0$ is on the plane wall. The equivalent $k$ values are 0.13 and 0.02 for $G/D=1.0$ and 2.0, respectively. A transition from $1/2$ to $8/16$ and then to QP is observed for $k=0.20$ and $f_{d}/f_{st\ast }$ in the range of 1.35–1.37 (as shown in figure 18a). The results shown in figure 18 suggest that the period doubling and transition to QP are indeed $k$-dependent. No period doubling occurs in the case with small $k=0.05$ in the uniform linear shear flow. The period doubling occurs in both uniform linear shear flow cases with $k>0.05$. Although period doubling from $1/2$ to $2/4$ occurs in reasonably wide bands of $f_{d}/f_{st\ast }$ for the cases with $k=0.10$ and 0.20, no transition to QP is observed within the Arnold tongue. The above observations appear to support our inference that the interaction of vortex shedding from the cylinder with the plane-wall boundary layer is primarily responsible for the transition to QP through cascade of period doublings inside the Arnold tongue. The transition to QP in the range of $1.35<f_{d}/f_{st\ast }<1.37$ mentioned above is outside of the Arnold tongue, which is different from that observed under the fixed-wall conditions. No further attempts are made to increase the $k$ value due to the numerical difficulty associated with possible negative velocity near the lower lateral boundary of the computational domain.

The occurrence of flow instabilities inside the otherwise synchronisation region has also been observed previously. For example, Ren et al. (Reference Ren, Cheng, Tong, Xiong and Chen2019) found regime D and F flows (spatio-temporal reflection symmetry breaking) within a region occupied by regime A flow (with spatio-temporal symmetry) at small gap ratios in oscillatory flow around four cylinders in a diamond arrangement. They attribute this strange behaviour to mode competition of the flows through the gaps between the cylinders and around the cylinder cluster. Similar mode completion mechanisms also exist in the cases studied in the present study. This interpretation is somewhat similar to the flow asymmetry discussed in the present study. Li et al. (Reference Li, Yao, Yang, Jaiman and Khoo2016) numerically investigated the VIV of a circular cylinder at $Re=200$. It was found that the VIV is periodic ($1/2$ regime) for an isolated cylinder with reduced velocity of 6.5 and 7, but it becomes quasi-periodic when a plane wall is introduced with $G/D=1.0$. It is speculated that the transition to QP of VIV in the case reported by Li et al. (Reference Li, Yao, Yang, Jaiman and Khoo2016) is also strongly related to the shear layer developed on the plane wall. No further investigation is carried out to substantiate this speculation because it is outside the scope of the present study.

4.1.5 Modes with $q>10$ and hysteresis

Tang et al. (Reference Tang, Cheng, Tong, Lu and Zhao2017) demonstrated that more synchronised modes with different $p/q$ ratios could be identified if the control parameters are further refined. This feature also applies to the near-wall cylinders that are investigated in this study. Table 3 provides a summary of 11 OMs with $10<q<20$ at $G/D=1.0$ under the moving-wall conditions. For example, a mode of $7/18$ is observed at $f_{d}/f_{St^{\ast }}=2.497$, which suggests that seven pairs of vortices are shed in $18T_{d}$. Figure 19 shows four instantaneous flow structures of mode $7/18$ over $18T_{d}$. The four inclined dashed lines illustrate a method to separate the groups of vortices. With these inclined dashed lines, we are able to separate the groups of seven pairs of vortices shed in 18 oscillation cycles (mode $7/18$). The vortex-shedding process repeats in every 18 cycles of cylinder oscillation. Therefore, by following the four inclined dashed lines, the vortex-shedding process in three successive 18 cylinder oscillation cycles can be observed. This process is essentially the same as the modes with $q$ lower than 10.

Figure 19.

Instantaneous vorticity contours of mode $7/18$ for an oscillating cylinder near a moving wall at $(f_{d}/f_{St^{\ast }},A/D,G/D)=(2.497,0.1,1)$. Four inclined dashed lines are used to separate the groups of seven pairs of vortices in $18T_{d}$.

Table 3.

OMs with $10<q<20$ at $G/D=1.0$ under moving-wall conditions.

Hysteresis is checked for three cases near a critical point $(f_{d}/f_{St^{\ast }},A/D,G/D)=(1.594,0.1,1.0)$ under the fixed-wall conditions, and the results are detailed in table 4. The initial conditions from a fully developed mode $1/2$ flow field at $(f_{d}/f_{St^{\ast }},A/D,G/D)=(1.594,0.1,1)$ are used to test the hysteresis in these cases. Hysteresis is only found for $f_{d}/f_{St^{\ast }}=1.584$, suggesting a very minor hysteresis effect in the wake of the cylinder under forced in-line oscillations based on the limited number of simulations. Further investigations with more simulations are beyond the scope of the present study.

Figure 20.

Effect of the oscillation amplitude on the synchronisation modes for a near-wall cylinder at $G/D=1.0$ under fixed-wall conditions. The shaded areas and dashed lines show the locations of selected modes. Refer to figure 5 caption for more details.

Table 4.

Synchronisation mode under different initial conditions.

4.2.1 Synchronisation modes

The influence of oscillation amplitude on the synchronisation modes in the $(f_{d}/f_{St^{\ast }},A/D)$ plane is illustrated in figure 20 for the fixed-wall conditions with $G/D=1.0$. A fine increment of 0.01 of $f_{d}/f_{St^{\ast }}$ is used for $A/D=0.2$ and 0.3 (except for mode $0/1$ at $A/D=0.3$, where an increment of 0.05 is adopted). Five $f_{d}/f_{St^{\ast }}$ ratios are selected (i.e. $f_{d}/f_{St^{\ast }}=1.250$, 1.563, 1.876, 2.188 and 2.501), and simulations are conducted with a fine $A/D$ increment of 0.01 within $A/D\in [0.01,0.4]$. To confirm the boundaries of modes $1/3$ and $0/1$, the flow with $f_{d}/f_{St^{\ast }}=2.897$ is simulated with $A/D\in [0.1,0.3]$, and two more cases with $f_{d}/f_{St^{\ast }}=3.438$ and 3.959 are simulated with $A/D\in [0.2,0.3]$. The shaded areas and dashed lines show the bands of synchronisations, which are referred to Arnold tongues according to Olinger & Sreenivasan (Reference Olinger and Sreenivasan1988).

Generally, three trends are observed on the synchronisation modes as $A/D$ increases. Firstly, the $f_{d}$ value that is required for the occurrence of a certain mode is reduced when $A/D$ increases, which is consistent with the results of an isolated cylinder as reported by Tang et al. (Reference Tang, Cheng, Tong, Lu and Zhao2017) and Leontini et al. (Reference Leontini, Lo Jacono and Thompson2013). Secondly, the number of synchronisation cases for certain dominant modes, such as modes $2/3$, $3/5$, $1/2$, $3/7$ and $1/3$, significantly decreases as $A/D$ increases from 0.1 to 0.2. Thirdly, the variety of synchronisation modes decreases with increasing $A/D$. For example, modes $6/7$, $5/6$, $4/5$, $3/4$ and $1/4$ are observed at $A/D=0.1$ but not at $A/D=0.2$. In addition, modes $2/3$, $3/5$, $3/7$, $3/8$ and $1/3$ are not found at $A/D=0.3$. Tang et al. (Reference Tang, Cheng, Tong, Lu and Zhao2017) pointed out that fine increments are required to systematically capture synchronisation modes composed of relatively large natural numbers of $p$ and $q$. For relatively small $f_{d}/f_{St^{\ast }}$, the reduction of synchronisation modes with increasing $A/D$ is again because the modes with very narrow bands of synchronisation require extremely fine increments of $f_{d}/f_{St^{\ast }}$. Moreover, at $A/D=0.3$ with relatively large $f_{d}/f_{St^{\ast }}$, a large area is covered by mode $0/1$, which indicates that the flow is dominated by the cylinder oscillation in this regime. The domination of cylinder oscillation is the reason why some of the synchronisation modes, such as modes $1/3$ and $1/4$, disappeared at this range of $A/D$ and $f_{d}/f_{St^{\ast }}$.

Two substantial differences are observed by comparing the present regime map with that for an isolated cylinder reported by Tang et al. (Reference Tang, Cheng, Tong, Lu and Zhao2017). Firstly, two large bands of modes $2/5$ and $1/3$ are observed for the near-wall cylinder, while the modes $2/5$ and $1/3$ bands are extremely narrow for an isolated cylinder. As discussed in the previous section, the flow asymmetry is responsible for the significant increase in synchronisation modes with an odd number as the denominator. Secondly, modes with reducible mode ratios (such as modes $2/4$, $4/6$ and $4/8$) are more pronounced at large $A/D$ values. Most of the modes with a reducible ratio (modes $2/4$ and $4/8$) are found within the range of the primary synchronisation regime, while a few are found outside this range. For example, a couple of mode $4/6$ cases are found near mode $2/3$. Furthermore, mode $2/6$ is found near mode $1/3$ at $(f_{d}/f_{St^{\ast }},A/D)=(2.501,0.14{-}0.16)$. The period doublings are more pronounced at larger $A/D$ values because the vortex shedding due to the cylinder oscillation is enhanced. Another obvious difference is the transition to QP within the boundary of mode $1/2$, which was not observed for an isolated cylinder (e.g. Leontini et al. Reference Leontini, Lo Jacono and Thompson2011, Reference Leontini, Lo Jacono and Thompson2013; Tang et al. Reference Tang, Cheng, Tong, Lu and Zhao2017).

4.2.2 Frequency content of the lift

The frequency of lift fluctuations is referred to as the fundamental frequency of vortex shedding in the literature (Williamson Reference Williamson1995). The fundamental frequency of lift is analysed here through fast Fourier transforms. The energy spectra of different frequency components are shown in figure 21 with greyscale contours. The spectra from each value of $f_{d}/f_{St^{\ast }}$ are stacked next to each other to show the frequency response of the lift for the near-wall cylinder as a function of $f_{d}/f_{St^{\ast }}$. The amplitude of each frequency component in the spectrum is normalised by the number of sampling points in each case. The cutoff ranges of the contours are selected to ensure that the frequency components with high energy are shown in the figure. The synchronisation modes are also labelled as a reference. In this work, the change in the vortex-shedding frequency against $f_{d}/f_{St^{\ast }}$ is illustrated at $G/D=1.0$ and three different $A/D$ ratios under the fixed-wall conditions.

Figure 21.

Contours of the frequency content of the lift signal as a function of $f_{d}/f_{St^{\ast }}$ when $G/D=1.0$ under fixed-wall conditions with greyscale contours: (a) $A/D=0.1$, (b) $A/D=0.2$ and (c) $A/D=0.3$. The maximum and minimum values of the greyscale shading are 0.2 and 0.002, respectively. Refer to figure 20 for the legend of synchronisation modes.

The strongest frequency component occurs at the fundamental vortex-shedding frequency, which is shown in darker colour in figure 21, accompanied by the sum and difference frequencies of the primary vortex-shedding frequency and driving frequency. A detailed discussion regarding the frequency variation for $G/D=\infty$ against $A/D$ was provided by Leontini et al. (Reference Leontini, Lo Jacono and Thompson2011), and the change in the vortex-shedding frequency as a function of $f_{d}$ at $A/D=0.1$ was provided by Tang et al. (Reference Tang, Cheng, Tong, Lu and Zhao2017). At $A/D=0.1$ (figure 21a), a gradual transition (reduction) process is observed for the dominating frequency ($f_{L}/f_{d}$) with increasing $f_{d}/f_{St^{\ast }}$. During this gradual transition stage, the flow mode $p/q$ changes from $1/1$ to $1/4$ (reduction). A plateau in the $f_{L}/f_{d}$ versus $f_{d}/f_{St^{\ast }}$ plot indicates synchronisation to a mode ratio. For example, the flow locks onto mode $1/2$ for a large range of $f_{d}/f_{St^{\ast }}$ (i.e. $f_{d}/f_{St^{\ast }}\in [1.59,2.06]$), onto mode $1/3$ for $f_{d}/f_{St^{\ast }}\in [2.64,3.01]$, and onto mode $1/4$ for $f_{d}/f_{St^{\ast }}\in [3.77,3.84]$.

Apart from the primary vortex-shedding frequency ($f_{s}$), other strong frequencies (light-grey lines) can be observed in figure 21. The first frequency is symmetrical about $f_{s}/f_{d}$ with respect to the line $f_{L}/f_{d}=0.5$, which is quantified as $f_{d}-f_{s}$. The second is $f_{d}$, a well-defined line of $f_{L}/f_{d}=1$. Similar to an isolated cylinder, all the frequency contents can be expressed as linear combinations of $f_{d}$ and $f_{s}$ as $|\pm \,Mf_{s}\pm Nf_{d}|$. The frequency contents in the range $f_{L}/f_{d}\in [1,2],[2,3]$ and so on are the scaled (down) images of $f_{L}/f_{d}\in [0,1]$ with weaker energy. The relationships of $f_{L}/f_{d}$ versus $f_{d}/f_{St^{\ast }}$ are rather similar for $A/D=0.1$ and $A/D=0.2$, except that the gradual transition occurs earlier for $A/D=0.2$. However, the variation in the frequency content of the lift becomes less obvious for $A/D=0.3$, and $f_{L}/f_{d}$ shows step changes when the regime transitions happen. This phenomenon occurs because the variation in synchronisation modes and the number of QP cases between these regimes significantly decrease due to the high level of disturbances. For mode $0/1$, $f_{L}/f_{d}\approx 1$ suggests that two pairs of vortices are shed at the same time during one period of cylinder oscillation. Moreover, the lowest frequency content observed at $A/D=0.2$ and $f_{d}/f_{St^{\ast }}>2.65$ (figure 21b) represents the secondary vortex-shedding frequency ($f_{SV}$) of the QP mode. Further analyses for mode $0/1$ and the secondary vortex shedding are detailed in § 4.2.3.

4.2.3 Mode $0/1$

For relatively large $A/D$ and $f_{d}/f_{St^{\ast }}$, a large area in the parameter space is covered by mode $0/1$. For an isolated cylinder with mode $0/1$, a pair of symmetric vortices forms in each cylinder oscillation period (resulting in zero lift), and the vortex shedding from a steady approaching flow is deemed to be fully suppressed (Tang et al. Reference Tang, Cheng, Tong, Lu and Zhao2017). The time sequence for the binary vortex street of an isolated cylinder can be found in figure 2 of Xu et al. (Reference Xu, Zhou and Wang2006). Although the symmetry in the wake is broken when a plane wall is introduced, the $0/1$ mode still exists.

A detailed analysis regarding the $0/1$ mode for $G/D=1.0$ is conducted here to understand the wall effect on this particular flow mode under the fixed-wall conditions. Three main features of the $0/1$ mode are discussed here:

1. (i) The boundary line for mode $0/1$ in figure 20 is plotted in figure 22, together with the data from Tang et al. (Reference Tang, Cheng, Tong, Lu and Zhao2017) and Leontini et al. (Reference Leontini, Lo Jacono and Thompson2013) for an isolated cylinder. The boundary line from Tang et al. (Reference Tang, Cheng, Tong, Lu and Zhao2017) was estimated from the five labelled data points. The comparison in figure 22 indicates that the fixed wall with $G/D=1.0$ does not significantly change the onset condition of mode $0/1$.

2. (ii) The lift is not zero for mode $0/1$ in the case of a fixed wall at $G/D=1.0$ due to the flow asymmetry with respect to $y=0$. The lift fluctuates periodically with the cylinder oscillations. A typical $0/1$ wake structure for $G/D=1.0$ is shown in figure 23(c), where vortices are shed simultaneously from the bottom and top sides of the cylinder and two nearly parallel shear layers form in the wake, similar to the results reported in Tang et al. (Reference Tang, Cheng, Tong, Lu and Zhao2017). The corresponding temporal variation of lift is shown in figure 24(cii). The Lissajous phase diagram of $X(\unicode[STIX]{x1D70F})$ and $C_{L}(\unicode[STIX]{x1D70F})$ shows a double-lobed structure. The Lissajous phase diagram for mode $0/1$ could also be a single loop, which is similar to that for mode $1/1$. For this reason, the Lissajous phase diagram alone is not sufficient to identify the synchronisation mode of the flow. Both flow patterns and the Lissajous phase diagram are used to identify flow synchronisation modes in this work.

3. (iii) A large area below the $0/1$ mode region in figure 20 is covered with the QP mode. A secondary wake is observed in the wake for cases with this QP mode. One typical example of the QP mode is shown in figures 23(b) and 24(b), $(f_{d}/f_{St^{\ast }},A/D)=(3.8,0.2)$. In the near wake $(x/D<4)$, the vortices that are shed from the cylinder surface are very similar to those observed in the $0/1$ mode (figure 23c), but further downstream ($x/D>5$), the small vortices that are shed from the cylinder merge into larger vortices and form a new secondary vortex street. The vortex-shedding frequency in the secondary wake is 0.1544, which is 22 % of $f_{d}$. The formation of this secondary vortex street is attributed to the instability of the shear layers in the wake. Similar observations were reported in the wake of a stationary cylinder (Thompson et al. Reference Thompson, Radi, Rao, Sheridan and Hourigan2014). When $A/D$ is further reduced to 0.1, the wake flow regains synchronisation to mode $1/4$, $(f_{d}/f_{St^{\ast }},A/D)=(3.8,0.1)$, as shown in figures 23(a) and 24(a). In the $1/4$ mode, the vortex shedding occurs alternatively in the wake near the cylinder, which is fundamentally different from the other two QP modes.

Figure 22.

Boundary line for mode $0/1$ extracted from figure 20 alongside the boundary line for an isolated cylinder from Tang et al. (Reference Tang, Cheng, Tong, Lu and Zhao2017) and a boundary point from Leontini et al. (Reference Leontini, Lo Jacono and Thompson2013).

Figure 23.

Instantaneous vorticity flow fields for a near-wall cylinder under fixed-wall conditions with $G/D=1.0$: (a) an example case of mode $1/4$, with $(f_{d}/f_{St^{\ast }},A/D)=(3.8,0.1)$; (b) a QP case with secondary vortex shedding (SVS) after binary vortices (BV), with $(f_{d}/f_{St^{\ast }},A/D)=(3.8,0.2)$; and (c) a case of mode $0/1$ with BV and parallel shear layers (PLS), with $(f_{d}/f_{St^{\ast }},A/D)=(3.8,0.3)$.

Figure 24.

The spectrum of $C_{L}$, the transient trace of $C_{L}(\unicode[STIX]{x1D70F})$ (red line) with cylinder displacement $X(\unicode[STIX]{x1D70F})$ (black line), and the Lissajous phase diagram of $X(\unicode[STIX]{x1D70F})$ and $C_{L}(\unicode[STIX]{x1D70F})$ for a near-wall cylinder with $G/D=1.0$ under the fixed-wall conditions: (a) mode $1/4$, $(f_{d}/f_{St^{\ast }},A/D)=(3.8,0.1)$; (b) QP, $(f_{d}/f_{St^{\ast }},A/D)=(3.8,0.2)$; and (c) mode $0/1$, $(f_{d}/f_{St^{\ast }},A/D)=(3.8,0.3)$.