3.1.1 Cylinder dynamics

Figure 4.

Maximum non-dimensional amplitude oscillation as a function of the reduced velocity for (a) positive values and (b) negative values of the rotating parameter  $\tilde{k}_{1}$ .

Owing to the character of the rotation imposed, symmetry is not broken and oscillations take place around the equilibrium position without flow ( $Y=0$ ) for all values of $U^{\ast }$ and $\tilde{k}_{1}$ . Thus, the cylinder’s response can be described by its main frequency of oscillation $f^{\ast }=f/f_{N}$ and the dimensionless amplitude $A^{\ast }=A_{10}/D$ of the cylinder’s steady-state oscillations.

Figure 4 shows the variation of the amplitude of oscillation for different values of $\tilde{k}_{1}$ . With respect to the non-rotating case (that is, $\tilde{k}_{1}=0$ ), positive values of $\tilde{k}_{1}$ lead to a reduction of the maximum amplitude of oscillation as well as a reduction of the region of $U^{\ast }$ where the cylinder undergoes large-amplitude oscillations. However, as can be seen, between $\tilde{k}_{1}=1.25$ and $\tilde{k}_{1}=2.5$ , the reduction in amplitude of oscillation is not very significant (figure 4 b). If $\tilde{k}_{1}$ is further increased, the picture does not vary much (regarding amplitude of oscillation, or synchronization region). With respect to the different branches of the curve, transition between the initial and upper branches is anticipated as $\tilde{k}_{1}$ increases. Finally, it can be noted that the differentiation between upper and lower branches is increasingly lost.

For negative values of $\tilde{k}_{1}$ , the picture is quite different (see figure 4 b): oscillations are significantly enhanced and two different responses can be observed. For $-1.8<\tilde{k}_{1}<0$ , the response amplitude exhibits in all cases a bell-shaped evolution as a function of the reduced velocity, which resembles the classical VIV response. This suggests that the vibrations occur under a wake–body synchronization mechanism. As $\tilde{k}_{1}$ becomes more negative, oscillations are more intense, the range of synchronization is enlarged, and the reduced velocity at which maximum amplitude is achieved is delayed as well as the initial–upper branch transition. A different behaviour has been found when $\tilde{k}_{1}<-1.8$ . As can be observed, for $\tilde{k}_{1}=-2.125$ , just after the initial–lower transition, $A^{\ast }$ seems to adopt a quasi-linear dependence on the reduced velocity. This resembles the dependence of galloping-type instability where motion-induced forces are dominant, in contrast to VIV where fluid forces are mainly driven by vortex shedding. For $\tilde{k}_{1}=-2.125$ , values up to $U^{\ast }=16$ only are presented. The reason has to do with the actual limitations of the experimental set-up. Specifically, beyond that threshold, the amplitude of the oscillations of the elastic blades is such that their elastic response starts to lose linearity. This raises the question of whether a galloping-like response will continue forever, or at some value of the reduced velocity the oscillations would start to decay. To analyse this point in more detail, an analysis of the near-wake flow pattern is made in the following subsection (figure 7), which shows that synchronization between vortex shedding and oscillations is lost. In addition, a quasi-steady model is developed in the next section, which explains the galloping-type response observed. The value $\tilde{k}_{1}=-1.875$ shows an intermediate behaviour: after the initial–upper branch transition, the amplitude response curve adopts a linear trend with the reduced velocity, but at $U^{\ast }\approx 10$ it loses this trend following an almost constant amplitude of oscillation for increasing $U^{\ast }$ until large-amplitude oscillations cease abruptly at  $U^{\ast }=20$ .

In figure 5, the normalized frequency of oscillations $f^{\ast }$ is plotted as a function of the reduced velocity for different values of $\tilde{k}_{1}$ . Owing to the large value of the reduced mass, variation of $f^{\ast }$ with $U^{\ast }$ is relatively small. In the initial branch the frequency is lowered slightly and then increases linearly with $U^{\ast }$ . For $\tilde{k}_{1}>0$ the slope of the frequency variation with $U^{\ast }$ is significantly increased. On the other hand, for negative values of $\tilde{k}_{1}$ (up to $\tilde{k}_{1}>-1.8$ ), $f^{\ast }$ is lowered and the slope with $U^{\ast }$ is smaller. Importantly, for large negative values of $\tilde{k}_{1}$ (i.e. $\tilde{k}_{1}<-1.8$ ), $f^{\ast }$ reaches a near-constant value after $U^{\ast }\approx 10$ . This result reinforces the idea that for large negative values of $\tilde{k}_{1}$ the cylinder undergoes galloping-type oscillations. Seemingly, the initial–upper branch transition branch occurs near $f^{\ast }=1$ for all values of $\tilde{k}_{1}$ (similar to the classical VIV response).

Figure 5.

Non-dimensional oscillating frequency of the cylinder as a function of the reduced velocity for (a) positive values and (b) negative values of the rotating parameter  $\tilde{k}_{1}$ .

3.1.2 PIV of the near wake for rotation proportional to the position

Here, PIV visualizations of the near wake are presented to understand the effect of the rotation of the cylinder on the wake patterns. Non-dimensional vorticity contours are given for different values of $\tilde{k}_{1}$ at two values of the reduced velocity, $U^{\ast }=5.2$ (near the peak amplitude) and $U^{\ast }=7.5$ , which correspond to the upper and lower branches, respectively, in the classical VIV curve ( $\tilde{k}_{1}=0$ ). The non-dimensional vorticity contour images have been treated by proper orthogonal decomposition (POD) so as to retain high-level energy modes and to filter high spatial and temporal frequencies (for further details, the reader is referred to Berkooz, Holmes & Lumley (Reference Berkooz, Holmes and Lumley1993) and Ma et al. (Reference Ma, Karniadakis, Park and Gharib2003)).

As mentioned earlier, the classical VIV response, for the Reynolds numbers under consideration ( $Re=3200$ ) and for low enough mass-damping parameter ( $m^{\ast }\unicode[STIX]{x1D701}=0.0506$ ), is characterized by a three-branch curve. Regarding the wake patterns, the initial branch is characterized by a 2S mode of vortex emission at the centreline. The upper branch presents a $2\text{P}_{0}$ mode of vortex shedding structure (as described in Morse & Williamson (Reference Morse and Williamson2009)), which consists of two pairs of vortices shed per cycle where the secondary vortex has a intensity of circulation smaller than the main vortex (figure 6 a). In the lower branch, the secondary vortex has grown in intensity developing into a full 2P mode (figure 6 b). Further increasing the velocity leads to a desynchronization between vortex emission and oscillation.

Figure 6.

Non-dimensional vorticity contours $\unicode[STIX]{x1D714}D/U$ for two values of reduced velocity ( $U^{\ast }=5.2$ , 7.5) and four values of $\tilde{k}_{1}$ ( $\tilde{k}_{1}=0$ , 1.25, $-1.0$ , $-1.875$ ), respectively. Red indicates clockwise circulation and blue indicates anticlockwise circulation. Vortical structures are highlighted with a red dashed line. P represents a pair of vortices shed per half-cycle corresponding to a 2P mode of vortex shedding. $\text{P}_{0}$ is equivalent to the previous one but with the secondary vortex being qualitatively smaller (circulation intensity-wise). S structures represent a single vortex being shed per half-cycle, which corresponds to a 2S mode of vortex shedding. Note that, for completeness, the amplitude of oscillation for each case is indicated by a double-headed arrow.

Figure 7.

Non-dimensional vorticity $\unicode[STIX]{x1D714}D/U$ contours for $U^{\ast }=17$ and $\tilde{k}_{1}=-1.875$ evaluated at four different displacements of the cylinder. Red indicates clockwise circulation and blue indicates anticlockwise circulation. The amplitude of oscillation is indicated by the double-headed arrow.

For $\tilde{k}_{1}>0$ , the wake structure in the initial branch is equivalent to the case without actuation (2S vortices shed near the centreline). After the transition, near the peak amplitude, the secondary vortex is reduced and even disappears as $\tilde{k}_{1}$ grows (figure 6 c), thus evolving from a $2\text{P}_{0}$ mode to a 2S mode of vortex shedding (as $\tilde{k}_{1}$ is increased) where two single vortices are shed per cycle at the outer part of the oscillation’s cycle (near its maximum amplitude). Also, the main vortex reduces its intensity compared to the case without actuation. This phenomenon probably occurs because the cylinder rotates in the opposite direction to the vortex being shed, therefore reducing the intensity of the circulation of the vortex shed, which contributes to a reduction of the amplitude of oscillation. As will be seen in the next subsection (§ 3.1.3), from a quasi-steady point of view, the lift coefficient depends on the rate of rotation, and positive values of $\tilde{k}_{1}$ lead to a lift coefficient opposite to the velocity of displacement, thus contributing to a reduction of the amplitude of oscillation expected. Finally, as $U^{\ast }$ grows, the upper and lower branches are no longer distinguishable and desynchronization is anticipated (figure 6 d).

For $\tilde{k}_{1}<0$ , the transition between the initial and upper branch is delayed; thus figure 6(e,g) shows a 2S vortex emission mode at the centreline, as these contours still represent the wake structure of the initial branch. As $U^{\ast }$ increases, after the transition, a $2\text{P}_{0}$ mode is obtained (or 2P mode). The vortices shed increase in strength (compared to $\tilde{k}_{1}=0$ ) as $\tilde{k}_{1}$ becomes more negative, because the rotation of the cylinder is in the same direction as that of the vortices being shed, thus increasing the circulation of such vortices (figure 6 f,h). The vorticity flux from the surface of the cylinder to the flow is $\unicode[STIX]{x1D6E4}_{s}=-\unicode[STIX]{x03C0}Du_{w}$ , where $u_{w}$ is the circumferential cylinder speed. For this rotating law, $u_{w}=Dk_{1}{\dot{y}}/2$ and therefore $\unicode[STIX]{x1D6E4}_{s}=-\unicode[STIX]{x03C0}D^{2}k_{1}{\dot{y}}/2$ , which suggests that vorticity emission from the cylinder’s surface contributes to increasing the circulation of the shed vortex when $k_{1}$ is negative. This contributes to the enhancement of the amplitude of oscillation. Noticeably, the vortices are shed at the outer part of the oscillation’s cycle; thus, it seems that the rotation of the cylinder (for the range of $\tilde{k}_{1}$ investigated) does not modify the phase at which such vortical structures are emitted.

The increase in amplitude is also related to the lift coefficient that appears due to the rotation of the cylinder (from a quasi-steady point of view) in the direction of the velocity of displacement, thus contributing to the enhancement of the oscillations for negative values of  $\tilde{k}_{1}$ .

Furthermore, a case for $U^{\ast }=17$ and $\tilde{k}_{1}=-1.875$ is analysed to observe the mechanism underlying oscillations for large negative values of $\tilde{k}_{1}$ and large $U^{\ast }$ . In figure 7, four different time snapshots are presented, and overprinted in a blue dashed line is the past trajectory of the cylinder. As can be observed, the picture is very different from the previous ones where there was synchronization between vortex emission and oscillation of the cylinder. In this case, the vortices are shed independently of the cylinder’s oscillation at its own frequency and thus the wake pattern resembles that of a wave as each vortex is shed independently at different points of the cycle of oscillation. In particular, the normalized vortex emission frequency is $f_{v}^{\ast }=f_{v}/f_{N}=3.44$ , which compares reasonably well to that of the Strouhal value of vortex emission for a cylinder at rest ( $St=0.21$ ), which yields $f_{vrest}^{\ast }=3.57$ ( $=StU^{\ast }$ ). Thus, for large values of negative $\tilde{k}_{1}$ and reduced velocity, there is no synchronization between vortex emission and oscillation, and the amplitude response tends to a linear dependence with $U^{\ast }$ resembling a galloping-type oscillation.

3.1.3 Quasi-steady model for galloping-type response

As observed in the previous subsection, there is a clear separation in the dynamical behaviour between the cases for $\tilde{k}_{1}>-1.8$ and those where $\tilde{k}_{1}<-1.8$ . In the latter case, from a certain value of the reduced velocity, the amplitude of oscillation increases linearly with the reduced velocity, leading one to assume that some kind of galloping-type phenomenon is happening. Also, the frequency of vortex emission for large $U^{\ast }$ follows the Strouhal law, therefore increasing linearly with $U^{\ast }$ , whereas the frequency of oscillation is locked at $f^{\ast }\approx 1$ . A quasi-steady model is proposed here to explain the observed behaviour, to determine for which values of $\tilde{k}_{1}$ a galloping-type behaviour is expected to hold, and to see the capability of the model to predict the cylinder’s dynamics by comparing the amplitude of oscillation given by the quasi-steady model with experimental results.

For the quasi-steady analysis, it is assumed that the instantaneous transverse force can be obtained from the steady lift and drag force with an equivalent instantaneous velocity and angle of attack $\unicode[STIX]{x1D6FC}_{F}$ (Blevins Reference Blevins1990). In figure 8(a) a scheme of the equivalent configuration is shown.

Figure 8.

(a) Schematics of the forces acting on the cylinder in a quasi-steady situation. (b) Average transverse force coefficient and average in-line force coefficient as a function of the dimensionless rotation rate $\unicode[STIX]{x1D6FC}$ . Experiments to determine $C_{L}$ and $C_{D}$ have been carried out at $Re\approx 9000$ , which corresponds to $U^{\ast }=15$ for the current set-up. Dashed lines represent the linear best fits.

Taking into account the lift and drag force coefficients and projecting onto the transverse (cross-flow) direction, one has

where $\unicode[STIX]{x1D6FC}_{F}$ is the instantaneous angle of attack, $\tan (\unicode[STIX]{x1D6FC}_{F})=-{\dot{y}}/U$ , $\cos (\unicode[STIX]{x1D6FC}_{F})=U/U_{r}$ , $U_{r}$ is the relative incident flow speed ( $U_{r}=\sqrt{U^{2}+{\dot{y}}^{2}}$ ), and $C_{L}$ and $C_{D}$ are, respectively, the lift and drag time-averaged fluid force coefficients, which can be obtained from the ‘static’ configuration as a function of the dimensionless rotation rate. Figure 8(b) shows the lift and drag coefficient variations with the dimensionless rotation rate obtained from experiments carried out in a water channel where the rotation rate was fixed at different values and fluid forces were measured using a force sensor. Assuming, as a first approximation, a linear variation with the rotation rate of $C_{L}$ and  $C_{D}$ , one has

When the cylinder rotates proportionally to the cylinder’s displacement ( $\unicode[STIX]{x1D703}^{\prime }=\tilde{k}_{1}Y^{\prime }$ ), (3.3a ) and (3.3b ) can be written in dimensionless form as where $U_{r}^{\ast }=U_{r}/f_{N}D=U^{\ast }/\cos (\unicode[STIX]{x1D6FC}_{F})$ is the non-dimensional relative incident flow velocity. From figure 8(b), the linear best fit yields $C_{L\unicode[STIX]{x1D6FC}}=-1.2$ , $C_{Do}=1.06$ and $C_{D\unicode[STIX]{x1D6FC}}=0.36$ , which are in good agreement when compared to similar experiments and numerical simulations (Mittal & Kumar Reference Mittal and Kumar2003; Karabelas et al. Reference Karabelas, Koumroglou, Argyropoulos and Markatos2012; Bourguet & Jacono Reference Bourguet and Jacono2014). Introducing (3.4a ) and (3.4b ) into (3.2), and taking into account that $\cos (\unicode[STIX]{x1D6FC}_{F})^{-1}=U_{r}/U$ , one obtains

which is the transverse fluid force coefficient evaluated from a quasi-steady basis. This model for the transverse force coefficient mainly neglects the effect of the vortices shed (note that there is no fluid force without cylinder motion), which is theoretically valid for $U^{\ast }$ large enough with respect to $St^{-1}$ (Parkinson Reference Parkinson1989). The first term in (3.5) can be seen as a positive linear damping for $\tilde{k}_{1}>0$ , reducing possible amplitude oscillations, or as a linear negative damping term for $\tilde{k}_{1}<0$ , which, on the other hand, contributes to amplifying the amplitude of oscillations. The last two terms are nonlinear damping terms associated with the drag force. Note that there is no stiffness term that could significantly change the frequency of oscillation, thus leading to the result that the frequency of oscillation should coincide with $f_{N}$ , that is, $f^{\ast }=1$ (which agrees well with results shown in figure 5(b) for $\tilde{k}_{1}=-1.875$ and $\tilde{k}_{1}=-2.125$ for large reduced velocities).

It is of interest to study the theoretical value of $\tilde{k}_{1}$ from which the galloping-type response is expected to occur according to the quasi-steady model. To do so, note that (3.5) can be linearized retaining only the linear terms:

Galloping-type oscillations may occur when the linearized term of the transverse force coefficients yields a positive value only, thus

which compares well with the results found during experimental results, since galloping-type oscillations appeared for $\tilde{k}_{1}<-1.8$ . To obtain the amplitude of oscillations, the dynamical equation for the cylinder (2.2a ) along with $C_{y}$ given by (3.5) can be solved analytically by the harmonic balance method such as in Vicente-Ludlam et al. (Reference Vicente-Ludlam, Barrero-Gil and Velazquez2014) and Vicente-Ludlam, Barrero-Gil & Velazquez (Reference Vicente-Ludlam, Barrero-Gil and Velazquez2015, Reference Vicente-Ludlam, Barrero-Gil and Velazquez2017) or numerically with a Runge–Kutta scheme.

Figure 9.

Comparison between quasi-steady solution and the present experimental results for $\tilde{k}_{1}=-2.125$ .

Figure 9 assesses the validity of the quasi-steady model proposed with the present experimental results for $\tilde{k}_{1}=-2.125$ with good agreement for high enough reduced velocity (say $U^{\ast }>8$ ) with errors smaller than 10 %. This reinforces the idea that for large negative values of $\tilde{k}_{1}$ and enough reduced velocities the cylinder is driven mainly for motion-induced fluid forces, which can be reasonably well described by the quasi-steady hypothesis. However, for $\tilde{k}_{1}=-1.875$ the quasi-steady model fails to predict the amplitude of oscillations of the experimental data since, after an initial linear trend with $U^{\ast }$ , the amplitude of oscillations keeps a nearly constant value until these cease at $U^{\ast }=20$ . This could be caused by the low value of the linear part of the transverse force from (3.5) at this value of $\tilde{k}_{1}$ , which is very close to the minimum theoretically predicted by the quasi-steady model, making other phenomena not taken into account in the quasi-steady model overcome the galloping-type oscillations.

3.2.1 Cylinder dynamics

Figure 10.

Maximum non-dimensional amplitude oscillation as a function of the reduced velocity for (a) positive values and (b) negative values of the rotating parameter  $\tilde{k}_{2}$ .

In figure 10, the amplitude of oscillation of the cylinder is plotted as a function of the reduced velocity for different values of $\tilde{k}_{2}$ . For $\tilde{k}_{2}>0$ the response is greatly attenuated with respect to the curve without actuation (figure 10 a, see curve $\tilde{k}_{2}=0$ ). With $\tilde{k}_{2}=1.25$ , the peak amplitude is $A^{\ast }\approx 0.22$ whereas when rotating proportionally to the cylinder’s displacement, for $\tilde{k}_{1}=2.5$ (which approximately leads to twice the level of rotation for a prescribed amplitude of oscillation) the maximum oscillation does not exceed $A^{\ast }=0.5$ (as was shown in figure 4 a). Thus rotating the cylinder proportionally to its velocity of oscillation is considerably more efficient in terms of attenuating the amplitude of oscillation with less actuation. On the other hand, negative values of $\tilde{k}_{2}$ lead to an increase of the amplitude of oscillation as well as an increase of the synchronization region undergoing large-amplitude oscillations. However, as can be seen in figure 10(b), further decreasing $\tilde{k}_{2}$ does not modify the type of response obtained, namely, no galloping-type oscillations have been found. This result can be explained through the quasi-steady model, which gives the following transverse fluid force coefficient:

This equation is obtained in the same way as (3.5), but taking into account that rotation is now proportional to the velocity of oscillation. From (3.9) note that it is not possible to introduce negative damping that can overcome the damping introduced by the drag coefficient. Thus, the increase of the amplitude of oscillations has to be explained by the modification of the structure of the near wake (namely, intensity of vortices shed). The transition between the initial and upper branch is not as pronounced as it is for the case without actuation (or when rotating the cylinder proportionally to its position), and the reduced velocity at which the transition occurs is not modified by $\tilde{k}_{2}$ . Also, the distinction between the upper and lower branches is lost as actuation (either positive or negative) is increased.

Finally, with regard to the frequency of oscillation (figure 11), $\tilde{k}_{2}$ does not modify $f^{\ast }$ significantly. The value of the non-dimensional frequency of oscillation nearly fits into a single curve for all values of $\tilde{k}_{2}$ , thus explaining why the initial–upper branch transition occurs at similar reduced velocity as the frequency of oscillation reaches $f^{\ast }=1$ at nearly the same reduced velocity. Negative values of $\tilde{k}_{2}$ (figure 11 b) increase the region of synchronization; therefore, the frequency of oscillation keeps growing with the same slope until desynchronization occurs (for larger $U^{\ast }$ as $\tilde{k}_{2}$ becomes more negative).

Figure 11.

Non-dimensional oscillating frequency of the cylinder as a function of the reduced velocity for (a) positive values and (b) negative values of the rotating parameter  $\tilde{k}_{2}$ .

3.2.2 PIV of the near wake for rotation proportional to the velocity of displacement

Figure 12.

Non-dimensional vorticity contours for two values of reduced velocity ( $U^{\ast }=5.5$ , 7.5) and four values of $\tilde{k}_{2}$ ( $\tilde{k}_{2}=0$ , 0.625, $-0.625$ , $-1.25$ ), respectively. Red indicates clockwise circulation and blue indicates anticlockwise circulation. Vortical structures are highlighted with a red dashed line. P represents a pair of vortices shed per half-cycle corresponding to a 2P mode of vortex shedding. $\text{P}_{0}$ is equivalent to the previous one but with the secondary vortex being qualitatively smaller (circulation intensity-wise). S structures represent a single vortex being shed per half-cycle, which corresponds to a 2S mode of vortex shedding. Note that, for completeness, the amplitude of oscillation for each case is indicated by a double-headed arrow.

Figure 13.

Non-dimensional vorticity contours for $U^{\ast }=9.5$ and three values of $\tilde{k}_{2}$ ( $\tilde{k}_{2}=0$ , $-0.625$ , $-1.25$ ), respectively. Red indicates clockwise circulation and blue indicates anticlockwise circulation. Vortical structures are highlighted with a red dashed line. P represents a pair of vortices shed per half-cycle corresponding to a 2P mode of vortex shedding. $\text{P}_{0}$ is equivalent to the previous one but with the secondary vortex being qualitatively smaller (circulation intensity-wise). S structures represent a single vortex being shed per half-cycle, which corresponds to a 2S mode of vortex shedding. Note that, for completeness, the amplitude of oscillation for each case is indicated by a double-headed arrow.

For positive values of $\tilde{k}_{2}$ , the initial branch is not modified with respect to the case without actuation (presenting a 2S mode of vortex shedding at the centreline). At  $U^{\ast }=5.5$ for $\tilde{k}_{2}=0$ (which corresponds to the upper branch), a $2\text{P}_{0}$ mode of vortex shedding was observed (figure 12 a). As $\tilde{k}_{2}$ increases, the amplitude of oscillation diminishes and still a $2\text{P}_{0}$ mode is observable (figure 12 c, $\tilde{k}_{2}=0.625$ ), though circulation intensity is reduced and vortices are shed nearer to the centreline. Further raising of $\tilde{k}_{2}$ leads to an increasingly undefined wake pattern where 2P vortex emission is mixed with 2S mode alternately (driven by the low amplitude of oscillation as well as the rotation of the cylinder). As the reduced velocity is increased ( $U^{\ast }=7.5$ ), the picture is similar where $2\text{P}$ vortex emission (for $\tilde{k}_{2}=0$ , figure 12 b) slowly evolves to a 2S mode as $\tilde{k}_{2}$ increases (figure 12 d exhibits a fully 2S mode for $\tilde{k}_{2}=0.625$ and amplitude of oscillation lower than 0.2).

For negative $\tilde{k}_{2}$ , the initial branch is not modified (2S mode). After the transition, in the upper branch ( $U^{\ast }=5.5$ ), negative $\tilde{k}_{2}$ markedly increases the circulation of the vortices shed compared to the case without actuation, and the secondary vortex is suppressed as $\tilde{k}_{2}$ is diminished (figure 12 e,g). For $U^{\ast }=7.5$ a 2P mode of vortex emission (similar to $\tilde{k}_{2}=0$ ) is obtained with increasing vortex circulation intensity as $\tilde{k}_{2}$ is diminished with also a diminishing intensity of the secondary vortex (figure 12 f,h). As observed, rotating the cylinder proportionally to its velocity mainly affects the intensity of the vortices shed for this range of $U^{\ast }$ without affecting the phase of vortex emission. For this rotating law, the vorticity flux from the surface of the cylinder is $\unicode[STIX]{x1D6E4}_{s}=-\unicode[STIX]{x03C0}D^{2}k_{2}\ddot{y} /2$ (note that circumferential velocity at the cylinder’s surface is $Dk_{2}\ddot{y} /2$ ). This suggests that more circulation in the shed vortex is expected for negative values of $k_{2}$ . Observe that if we consider $\ddot{y} \approx -2\unicode[STIX]{x03C0}f^{2}y$ (since oscillations have a harmonic character), it follows that $\unicode[STIX]{x1D6E4}_{s}$ is in phase with the cylinder’s displacement. This may help to accommodate vorticity generation in the surface onto the growing vortex, increasing its circulation significantly.

Notably, as seen in figure 10(b), the synchronization region can be enlarged with negative $\tilde{k}_{2}$ . In particular, non-dimensional vorticity contour maps are given at $U^{\ast }=9.5$ for the case without actuation ( $\tilde{k}_{2}=0$ , figure 13 a) and two negative values of $\tilde{k}_{2}$ ( $\tilde{k}_{2}=-0.625$ , $\tilde{k}_{2}=-1.25$ , figures 13 b and 13 c, respectively). As observed, for $\tilde{k}_{2}=0$ , oscillations are small due to desynchronization between vortex emission and the cylinder’s oscillation. However, as $\tilde{k}_{2}$ becomes more negative, the synchronization region is enlarged and large-amplitude oscillations occur and a fully developed 2P mode can be observed.

Finally, no quasi-steady flow pattern has been found with regard to the wake pattern (as deduced from (3.9)).