3.1 Control imposed with a rotation phase of $0^{\circ }$

To develop an understanding of the potential effect of imposed rotation on the vibration response of a sphere, it is worthwhile to first consider the isolated contribution of the Magnus effect induced by the imposed rotation over a vibration cycle. Put succinctly, the Magnus effect results in a lift force perpendicular to the axis of rotation, in the direction of the side of the sphere rotating with the free-stream flow (retreating side). The effect occurs due to the addition of momentum near the surface of the sphere, reducing the adverse pressure gradient on the retreating side which delays separation, whilst conversely increasing the adverse pressure gradient on the advancing side which promotes separation. The resultant pressure difference over the sphere produces a force in proportion to the Reynolds number and rotation ratio, $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FA}D/(2U)$, with $\unicode[STIX]{x1D6FA}$ the angular velocity, at least for small to moderate $\unicode[STIX]{x1D6FC}$. For certain flow conditions ($Re\sim 10^{5}$), a force in the opposite direction is generated; this is referred to as the ‘negative’ or ‘inverse’ Magnus effect. It is not expected that the inverse Magnus effect would be encountered over the Reynolds number range investigated in this study (Kim et al. Reference Kim, Choi, Park and Yoo2014). Indeed, even for the normal Magnus effect, there has been little research conducted on it over the Reynolds number range of the current study. Notably, Tsuji, Morikawa & Mizuno (Reference Tsuji, Morikawa and Mizuno1985) collated past results in the range $Re<1$ to $Re=1.1\times 10^{5}$ and conducted their own experiments over the range $5.50\times 10^{2}\leqslant Re\leqslant 1.60\times 10^{3}$. In addition, various numerical studies (e.g. Giacobello, Ooi & Balachandar Reference Giacobello, Ooi and Balachandar2009; Poon et al. Reference Poon, Ooi, Giacobello and Cohen2010; Dobson, Ooi & Poon Reference Dobson, Ooi and Poon2014; Rajamuni et al. Reference Rajamuni, Thompson and Hourigan2018b) have documented the effect for $Re\leqslant 1000$. Over this relatively low Reynolds number range of the current study, and the range of rotation ratios implemented, it is expected that the lift force due to the Magnus effect will increase monotonically with increasing rotation ratio. As a result, for rotation at $\unicode[STIX]{x1D719}_{rot}=0^{\circ }$, the force from the Magnus effect is expected to be $90^{\circ }$ out of phase with sphere transverse displacement, with the peak force occurring at $y/D=0$ in the direction opposite to the motion of the sphere. Therefore, if the Magnus effect were to act in isolation, it is expected that the amplitude of sphere vibration would be reduced for $\unicode[STIX]{x1D719}_{rot}=0^{\circ }$.

Figure 4 shows key characteristics of the vibration response of the sphere for the four proportional gain values investigated at $\unicode[STIX]{x1D719}_{rot}=0^{\circ }$. Commencing with the mode I response, it is apparent from figure 4(a) that the imposed rotation alters the reduced velocity at which the vibration is instigated. As previously discussed, it is expected that the Magnus effect due to the imposed rotation would suppress vibration. However, increasing the proportional gain results in a reduction in the reduced velocity at which the vibration is instigated, down to $U^{\ast }=4$ for $K_{p}^{\ast }=4$, the largest proportional gain tested. Over this regime, the vibrations occur close to the natural frequency of the system (figure 4b), there is a sharp increase in the root-mean-square (r.m.s.) transverse force coefficient, $C_{y}$, where the vibration is instigated, as for the natural response (figure 4c), and the total phase remains close to that observed for the natural response (figure 4d).

Figure 4.

Response of the sphere with imposed rotation at $\unicode[STIX]{x1D719}_{rot}=0^{\circ }$ as a function of $U^{\ast }$ for $K_{p}^{\ast }=0.5,1,2$ and 4. (a) Variation of vibration amplitude, $A^{\ast }$. (b) Dominant vibration frequency obtained from the power spectrum of transverse sphere position. (c) Variation of the r.m.s. transverse force coefficient, $C_{y}$. (d) Variation of total phase, $\unicode[STIX]{x1D719}_{total}$. (○, blue) $K_{p}^{\ast }=0$, (○, black) $K_{p}^{\ast }=4$, (▫) $K_{p}^{\ast }=2$, (▵) $K_{p}^{\ast }=1$, (♢) $K_{p}^{\ast }=0.5$. Note that $\unicode[STIX]{x1D719}_{total}$ is presented in degrees.

Figure 5 shows sample time traces of the transverse displacement and force coefficient at $U^{\ast }=4$ for the natural response and at $U^{\ast }=5.25$ with imposed rotation ($K_{p}^{\ast }=4$). As observed for the natural response in the mode I regime, with imposed rotation, the fluid force acting on the sphere is highly periodic and it is evident that the total phase is close to $0^{\circ }$. The time trace and power spectral density (PSD) plots of the transverse force coefficient from figure 5 highlight the similarity between the two responses, suggesting the imposed rotation promotes early onset of the mode I response as opposed to a new mode of vibration. Furthermore, if the vibration was instigated due to a Magnus force like effect, then it would be expected that, as a minimum, an inflection in the trace of transverse force coefficient (figure 5c) would be observed at $y/D=0$ where the Magnus effect is expected to be maximum. Rajamuni et al. (Reference Rajamuni, Thompson and Hourigan2018b) imposed constant rotation to a sphere constrained to one degree of freedom. They found that with the leading side of the sphere, in the transverse sense, rotating towards the oncoming free stream, the vortex street that developed either diminished in strength or was suppressed completely. Therefore, in this study, where for $\unicode[STIX]{x1D719}_{rot}=0^{\circ }$ the leading side of the sphere is always rotating towards the oncoming free stream, it was expected that the strength of the lift-inducing streamwise vortex structures would reduce. Rather, it is suspected that as the imposed rotation is linked to the system dynamics, periodic vortex shedding is promoted, enabling earlier onset and expansion of the resonance vibration regime.

Figure 5.

Variation in the characteristics of $C_{y}$ with imposed rotation in the mode I regime. (a,b) Natural response at $U^{\ast }=4$. (c,d) Response with $K_{p}^{\ast }=4$ and $\unicode[STIX]{x1D719}_{rot}=0^{\circ }$ at $U^{\ast }=5.25$. (a,c) Time traces of sphere displacement (black line) and $C_{y}$ (orange line). The horizontal axis shows time non-dimensionalised by the oscillation period, $\unicode[STIX]{x1D70F}=t/T$. (b,d) Power spectrum of $C_{y}$ as a function of $f^{\ast }$. Black dashed lines represent the vibration frequency of the sphere and higher harmonics. Note that for (a,c) the scale of $y/D$ and $C_{y}$ has been adjusted for each plot to aid comparison.

At higher reduced velocities, perhaps the most notable feature of the amplitude response shown in figure 4(a), is that the imposed rotation results in an attenuation of the maximum vibration amplitude. For the proportional gain values investigated, a maximum reduction in the peak vibration amplitude of 44 % was observed. It is also evident that the peak amplitude response occurs at lower reduced velocities with increasing proportional gain and that higher proportional gain values are associated with a broader peak vibration region. Govardhan & Williamson (Reference Govardhan and Williamson2005) showed that the mode II regime is associated with the total phase transitioning through $90^{\circ }$. As can be seen in figure 4(d), the imposed rotation causes the total phase to transition from close to $0^{\circ }$ through $90^{\circ }$ at a lower reduced velocity. This, along with the decreased reduced velocity at peak vibration, implies that the mode II regime is shifted to increasingly lower reduced velocities with increasing proportional gain. Interestingly, the vibration frequency remains at $f^{\ast }\approx 1.03$ at the peak of the mode II response for all proportional gain values investigated (figure 4b).

For the sphere to undergo sustained vibration, there must be a net energy transfer from the fluid to the sphere over a vibration cycle. A net energy transfer can only occur with $0^{\circ }<\unicode[STIX]{x1D719}_{total}<180^{\circ }$. For a given transverse force coefficient, the maximum energy transfer occurs with $\unicode[STIX]{x1D719}_{total}=90^{\circ }$, and reduces to zero at $\unicode[STIX]{x1D719}_{total}=0^{\circ }$ and $180^{\circ }$. So increasing the proportional gain, which causes the total phase to transition closer to $\unicode[STIX]{x1D719}_{total}=180^{\circ }$ in the mode II regime, reduces the potential for net energy transfer to the sphere.

Over the range $4\lesssim U^{\ast }\lesssim 12$, the imposed rotation noticeably alters the transverse force coefficient (figure 4c). The minimum in the transverse force coefficient, which is observed close to the peak of the mode II regime, remains almost constant for all proportional gain values. Past the peak of the mode II regime, there is a second maximum in the transverse force coefficient seen. The magnitude of the second maximum increases significantly with proportional gain. Considering the magnitude of the transverse force coefficient in conjunction with total phase, it is evident that the net energy transfer to the sphere remains relatively constant over a broad range of reduced velocities. This in turn may explain the relatively constant, and increasingly broad, peak vibration region seen with increasing proportional gain. Figure 6, which shows time traces of the transverse displacement and force coefficient at $U^{\ast }=10$, for the natural response and with imposed rotation ($K_{p}^{\ast }=4$), highlights how the imposed rotation continues to promote a highly periodic transverse force response past the mode I regime.

Figure 6.

Variation in the characteristics of $C_{y}$ with imposed rotation in the mode II regime ($U^{\ast }=10$). (a,b) Natural response. (c,d) Response with $K_{p}^{\ast }=4$ and $\unicode[STIX]{x1D719}_{rot}=0^{\circ }$. (a,c) Time traces of sphere displacement (black line) and $C_{y}$ (orange line). The horizontal axis shows time non-dimensionalised by the oscillation period, $\unicode[STIX]{x1D70F}=t/T$. (b,d) Power spectrum of $C_{y}$ as a function of $f^{\ast }$. Black dashed lines represent the vibration frequency of the sphere and higher harmonics. Note that for (a,c) the scale of $y/D$ and $C_{y}$ has been adjusted for each plot to aid comparison.

At $U^{\ast }\approx 12$, there is a distinct change in the response of the sphere for the largest proportional gain, $K_{p}^{\ast }=4$, which is visible across the range of vibration characteristics examined in figure 4. Here, the steepest decline in vibration amplitude is observed (figure 4a), there is a step change in the vibration frequency (figure 4b), a rapid reduction in the transverse force coefficient (figure 4c) and, lastly, an abrupt reduction in total phase (figure 4d). It is conjectured that the periodicity promoted by the imposed rotation can no longer be sustained as the equivalent fixed-body (i.e. stationary sphere) vortex shedding frequency increases, moving further from the natural vibration frequency of the system. Consequently, the Magnus effect begins to dominate and the amplitude response is further reduced. In figure 4(d) it can be seen that total phase falls close to $90^{\circ }$ at $U^{\ast }\approx 12$, and that it slowly decreases further at a similar rate to the natural response. In stark contrast to figures 5 and 6, figure 7 reveals a significantly decreased periodicity with imposed rotation.

Further to the reduction in periodicity of the transverse force as reduced velocity increases, a similar reduction is seen in the displacement response. To quantify this change, the definition of periodicity used by (Jauvtis et al. Reference Jauvtis, Govardhan and Williamson2001) is adopted here. Periodicity is defined as ${\mathcal{P}}=\sqrt{2}y_{rms}/y_{max}$, noting it attains a value of unity for a purely sinusoidal signal and reduces in value for a less periodic signal. Figure 8 shows a reduction in the periodicity of the sphere displacement with imposed rotation at $\unicode[STIX]{x1D719}_{rot}=0^{\circ }$. When proportional gain is set to larger values, the magnitude of this reduction is increased and it begins to occur at lower reduced velocities.

Figure 7.

Variation in the characteristics of $C_{y}$ with imposed rotation in the mode III transition regime ($U^{\ast }=13$). (a,b) Natural response. (c,d) Response with $K_{p}^{\ast }=4$ and $\unicode[STIX]{x1D719}_{rot}=0^{\circ }$. (a,c) Time traces of sphere displacement (black line) and $C_{y}$ (orange line). The horizontal axis shows time non-dimensionalised by the oscillation period, $\unicode[STIX]{x1D70F}=t/T$. (b,d) Power spectrum of $C_{y}$ as a function of $f^{\ast }$. Black dashed lines represent the vibration frequency of the sphere and higher harmonics. Note that for (a,c) the scale of $y/D$ and $C_{y}$ has been adjusted for each plot to aid comparison.

Figure 8.

Periodicity of the sphere displacement as a function of $U^{\ast }$. (○, blue) $K_{p}^{\ast }=0$, (○, black) $K_{p}^{\ast }=4$, (▫) $K_{p}^{\ast }=2$, (▵) $K_{p}^{\ast }=1$, (♢) $K_{p}^{\ast }=0.5$. The dashed line represents a purely sinusoidal signal.

In an effort to compare the magnitude of the imposed rotation to the open-loop control studies of Sareen et al. (Reference Sareen, Zhao, Lo Jacono, Sheridan, Hourigan and Thompson2018a,Reference Sareen, Zhao, Sheridan, Hourigan and Thompsonb), the mean effective velocity ratio, $\unicode[STIX]{x1D6FC}_{r}^{\ast }=\unicode[STIX]{x1D6FA}_{max}D/(2U)$, where $\unicode[STIX]{x1D6FA}_{max}$ is the maximum angular velocity over a vibration cycle, was calculated. As can be observed from figure 9, a significant benefit of the feedback control is that as the amplitude of imposed rotation is linked to the sphere displacement; when the vibrations are suppressed, the amplitude of the imposed rotation reduces in proportion. Consequently, past the peak of mode II, where the vibrations are significantly suppressed, the largest proportional gain value investigated, $K_{p}^{\ast }=4$, has a very small effective velocity ratio ($\unicode[STIX]{x1D6FC}_{r}^{\ast }<0.1$).

Figure 9.

Mean velocity ratio, $\unicode[STIX]{x1D6FC}_{r}^{\ast }$, as a function of $U^{\ast }$. (○) $K_{p}^{\ast }=4$, (▫) $K_{p}^{\ast }=2$, (▵) $K_{p}^{\ast }=1$, (♢) $K_{p}^{\ast }=0.5$.

3.2 Control imposed with a rotation phase of $180^{\circ }$

As may be expected, the amplitude response for $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$ is loosely inverse to that for $\unicode[STIX]{x1D719}_{rot}=0^{\circ }$. Overall, there is a slightly delayed instigation of the vibration for large proportional gains, an attenuation of vibration in the mode I regime and a monotonic increase from near the peak of the mode II regime onwards.

Figure 10.

Response of the sphere with imposed rotation at $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$ as a function of $U^{\ast }$ for $K_{p}^{\ast }=0.5,1,2$ and $4$. (a) Variation of vibration amplitude, $A^{\ast }$. (b) Dominant vibration frequency obtained from the power spectrum of transverse sphere position. (c) Variation of the r.m.s. transverse force coefficient, $C_{y}$. (d) Variation of total phase, $\unicode[STIX]{x1D719}_{total}$. (○, blue) $K_{p}^{\ast }=0$, (○, black) $K_{p}^{\ast }=4$, (▫) $K_{p}^{\ast }=2$, (▵) $K_{p}^{\ast }=1$, (♢) $K_{p}^{\ast }=0.5$. Note that $\unicode[STIX]{x1D719}_{total}$ is presented in degrees.

Figure 10 shows key characteristics of the vibration response of the sphere with $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$ for the four proportional gain values investigated. The amplitude response of the sphere is shown in figure 10(a). For the natural response, the vibration commences suddenly at $U^{\ast }=5.25$, with the amplitude of vibration increasing from $A^{\ast }=0$ to $A^{\ast }=0.37$. With increasing proportional gain, the magnitude of the initial vibration is increasingly attenuated. For the highest proportional gain tested, $K_{p}^{\ast }=4$, the vibration response increases slowly, and relatively linearly, up to $U^{\ast }=8.5$. For $U^{\ast }\lesssim 9$, the total phase remains close to $0^{\circ }$ for all proportional gain values (figure 10d), although there is a significant variation in the transverse force coefficient as shown in figure 10(c). Over this regime, as shown in figure 10(b), the vibration frequency is reduced, particularly for $K_{p}^{\ast }=4$.

As discussed in § 3.1, at low reduced velocities it appears that the imposed rotation promotes a highly periodic transverse force signal. For $\unicode[STIX]{x1D719}_{rot}=0^{\circ }$, this in turn resulted in highly periodic vibration. However, it was noted that the vibration was not anticipated given the expected direction of the Magnus effect. Here, for $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$, it was expected that the Magnus effect would contribute to increased vibration. Evidently, this did not occur. As can be seen from figure 11, the imposed rotation results in a highly periodic, albeit attenuated, transverse force signal. The combination of attenuated transverse force and shift in total phase closer to $0^{\circ }$, results in a reduction in the net energy transfer to the sphere over a vibration cycle in the mode I regime.

Figure 11.

Variation in the characteristics of $C_{y}$ with imposed rotation in the mode I regime ($U^{\ast }=6$). (a,b) Natural response. (c,d) Response with $K_{p}^{\ast }=4$ and $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$. (a,c) Time traces of sphere displacement (black line) and $C_{y}$ (orange line). The horizontal axis shows time non-dimensionalised by the oscillation period, $\unicode[STIX]{x1D70F}=t/T$. (b,d) Power spectrum of $C_{y}$ as a function of $f^{\ast }$. Black dashed lines represent the vibration frequency of the sphere and higher harmonics. Note that for (a,c) the scale of $y/D$ and $C_{y}$ has been adjusted for each plot to aid comparison.

Whereas for $\unicode[STIX]{x1D719}_{rot}=0^{\circ }$ and $K_{p}^{\ast }=4$, $U^{\ast }\approx 12$ marked a change in the characteristics of the vibration response, for $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$, $U^{\ast }\approx 9$ marks a distinct change in the response for all proportional gain values. At $U^{\ast }\approx 9$, prior to the peak of the mode II response seen with no imposed rotation, the amplitude of vibration for all proportional gain values investigated switches from an attenuated to an amplified response. Here, $U^{\ast }\approx 9$ coincides with the sharp increase in total phase seen for the natural response. From $U^{\ast }\approx 9$, for $K_{p}^{\ast }\geqslant 1$, the vibration response increases monotonically well above the amplitude observed with no imposed rotation. A similar trend was observed for a sphere (Sareen et al. Reference Sareen, Zhao, Sheridan, Hourigan and Thompson2018b) and circular cylinder (Zhao et al. Reference Zhao, Lo Jacono, Sheridan, Hourigan and Thompson2018) with imposed open-loop rotary oscillations, and for a circular cylinder with feedback control (Vicente-Ludlam et al. Reference Vicente-Ludlam, Barrero-Gil and Velazquez2018). This type of response has been labelled as ‘rotation-induced’ vibration, a result of the imposed rotation dominating the natural vortex shedding, causing a ‘galloping-like’ response. It is evident from figure 12 that over the rotation-induced vibration regime, the periodicity of the vibration approaches unity as proportional gain is increased.

Figure 12.

Periodicity of the sphere displacement as a function of $U^{\ast }$. (○, blue) $K_{p}^{\ast }=0$, (○, black) $K_{p}^{\ast }=4$, (▫) $K_{p}^{\ast }=2$, (▵) $K_{p}^{\ast }=1$, (♢) $K_{p}^{\ast }=0.5$. The dashed line represents a purely sinusoidal signal.

As shown in figure 13, for $U^{\ast }\lesssim 9$, the frequency spectrum of transverse force consists of a single distinct peak at $f_{C_{y}}^{\ast }=1$. Above $U^{\ast }\approx 9$, harmonics of $f^{\ast }$ begin to appear, and by $U^{\ast }=20$ the spectrum is dominated by vibration at $f^{\ast }=3$. Figure 14 highlights how the frequency response of the transverse force coefficient is dominated by the second harmonic of vibration frequency at $U^{\ast }=20$.

Figure 13.

Frequency PSD contour plots of $C_{y}$ against $U^{\ast }$. (a) Natural response; (b) $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$ and $K_{p}^{\ast }=4$. The spectral power is normalised by the maximum value at each $U^{\ast }$ and is presented on a $\log _{10}$ scale.

Figure 14.

Variation in the characteristics of $C_{y}$ with imposed rotation in the mode III transition regime ($U^{\ast }=20$). (a,b) Natural response. (c,d) Response with $K_{p}^{\ast }=4$ and $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$. (a,c) Time traces of sphere displacement (black line) and $C_{y}$ (orange line). The horizontal axis shows time non-dimensionalised by the oscillation period, $\unicode[STIX]{x1D70F}=t/T$. (b,d) Power spectrum of $C_{y}$ as a function of $f^{\ast }$. Black dashed lines represent the vibration frequency of the sphere and higher harmonics. Note that for (a,c) the scale of $y/D$ and $C_{y}$ has been adjusted for each plot to aid comparison.

As can be seen from figure 10(b), there is minimal variation in the vibration frequency for $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$. Vicente-Ludlam et al. (Reference Vicente-Ludlam, Barrero-Gil and Velazquez2018) reported that for the circular cylinder, the imposed feedback control resulted in the vibration frequency remaining constant close to $f^{\ast }=1$ in the rotation-induced vibration regime. Likewise, for $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$, above $U^{\ast }\approx 9$ where rotation-induced vibration is observed, the vibration frequency remained close to constant, slightly below that observed for the natural response. Over this regime, where the vibration response was amplified, the total phase progressively deviates from the trend seen for the natural response as proportional gain increases (figure 10d). For $K_{p}^{\ast }=4$, the total phase increases monotonically up to $\unicode[STIX]{x1D719}_{total}=76^{\circ }$ at $U^{\ast }=20$.

Once again, comparing the velocity ratio values to those for open-loop rotary control (Sareen et al. Reference Sareen, Zhao, Lo Jacono, Sheridan, Hourigan and Thompson2018a), we find that the feedback control provides a more efficient means to amplify the vibration (figure 15). Whilst not directly attempting to optimise the amplification of vibration, Sareen et al. (Reference Sareen, Zhao, Lo Jacono, Sheridan, Hourigan and Thompson2018a) did present results with imposed rotary oscillations at a frequency 1.1 times the natural frequency of the system with $\unicode[STIX]{x1D6FC}_{r}^{\ast }=1$, resulting in a vibration amplitude of $A^{\ast }=1$ at $U^{\ast }=20$. Here, we have two comparable data points at $U^{\ast }=20$; imposed rotation with $K_{p}^{\ast }=2$ and $\unicode[STIX]{x1D6FC}_{r}^{\ast }=0.51$ resulting in $A^{\ast }=1.27$, and imposed rotation with $K_{p}^{\ast }=4$ and $\unicode[STIX]{x1D6FC}_{r}^{\ast }=1.30$ resulting in $A^{\ast }=2.1$.

Figure 15.

Mean amplitude ratio, $\unicode[STIX]{x1D6FC}_{r}^{\ast }$, as a function of $U^{\ast }$. (○) $K_{p}^{\ast }=4$, (▫) $K_{p}^{\ast }=2$, (▵) $K_{p}^{\ast }=1$, (♢) $K_{p}^{\ast }=0.5$.

3.3 Effect of phase variation

In §§ 3.1 and 3.2, the effect of rotation at $\unicode[STIX]{x1D719}_{rot}=0^{\circ }$ and $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$ was examined. It was shown that varying the rotation phase significantly altered the vibration response of the sphere. To determine how the vibration response evolves over the range $0^{\circ }\leqslant$$\unicode[STIX]{x1D719}_{rot}\leqslant 360^{\circ }$, in this section the second control law (2.5) is implemented. In this study, the rotation phase is varied in $15^{\circ }$ increments for $K_{p}^{\ast }=0.5,1,2$ and 4 at reduced velocities of $U^{\ast }=6,10$ and 14 corresponding to mode I, mode II and mode III transition regimes for the non-rotating sphere.

Figure 16.

Response of the sphere in the mode I regime ($U^{\ast }=6$) with imposed rotation over the range $0^{\circ }\leqslant$$\unicode[STIX]{x1D719}_{rot}\leqslant 360^{\circ }$ (polar angle) for four proportional gain values $K_{p}^{\ast }=0.5,1,2$ and 4. Radius shows (a) $A^{\ast }$, (b) $f^{\ast }$, (c) $C_{y}$ and (d) $\unicode[STIX]{x1D719}_{total}$. In (a–d) (●) $K_{p}^{\ast }=4$, (▪) $K_{p}^{\ast }=2$, (▴) $K_{p}^{\ast }=1$, (♦) $K_{p}^{\ast }=0.5$. (e,f) PSD contour plots of $f^{\ast }$ and $f_{C_{y}}^{\ast }$, respectively, for $K_{p}^{\ast }=2$. The spectral power is normalised by the maximum value at each $\unicode[STIX]{x1D719}_{rot}$ and is presented on a $\log _{10}$ scale. The black dashed line represents the natural response.

3.3.2 Mode II

Previous studies implementing open-loop control for a sphere undergoing VIV have found it most difficult to alter the vibration response at the peak of the mode II regime (Sareen et al. Reference Sareen, Zhao, Lo Jacono, Sheridan, Hourigan and Thompson2018a,Reference Sareen, Zhao, Sheridan, Hourigan and Thompsonb). Figure 17 shows the response of the sphere with imposed rotation in the mode II regime ($U^{\ast }=10$). The imposed rotation results in a slight increase in the vibration amplitude, for all proportional gain values, over the range $165^{\circ }\lesssim \unicode[STIX]{x1D719}_{rot}\lesssim 300^{\circ }$ (figure 17a). Conversely, the vibrations are attenuated between $315^{\circ }\lesssim$$\unicode[STIX]{x1D719}_{rot}\lesssim 135^{\circ }$. The maximum attenuation for all proportional gain values is observed at $\unicode[STIX]{x1D719}_{rot}\approx 60^{\circ }$. It is not immediately obvious why this maximum attenuation occurs at this phase angle. Here, the transverse force is only slightly attenuated over the region $45^{\circ }\lesssim$$\unicode[STIX]{x1D719}_{rot}\lesssim 120^{\circ }$ (figure 17c). Additionally, there is no clear trend in total phase suggesting a particular amplitude response (figure 17d). At $\unicode[STIX]{x1D719}_{rot}\approx 60^{\circ }$, for $K_{p}^{\ast }\geqslant 2$, the total phase is reduced well below that for the natural response, whilst for $K_{p}^{\ast }\leqslant 1$, total phase is increased.

As can be seen from figure 17(e), for $K_{p}^{\ast }=4$, where the vibrations are amplified, the spectral power is concentrated at $f^{\ast }=1$. Whilst, where the vibrations are attenuated, the spectral power is distributed broadly over the range $0\lesssim f^{\ast }\lesssim 2$. Similarly, distinct differences in the frequency spectra of transverse force signals for the regimes of vibration amplification and attenuation can be observed in figure 17(f). Interestingly, where the vibrations are attenuated, there is a distinct lack of spectral power at $f_{C_{y}}^{\ast }=1$. Conversely, where the vibration is amplified, there is a strong peak in spectral power at $f_{C_{y}}^{\ast }=1$ and to a lesser extent, peaks at $f_{C_{y}}^{\ast }=2$ and $f_{C_{y}}^{\ast }=3$.

Figure 17.

Response of the sphere in the mode II regime ($U^{\ast }=10$) with imposed rotation over the range $0^{\circ }\leqslant \unicode[STIX]{x1D719}_{rot}\leqslant 360^{\circ }$ for four proportional gain values $K_{p}^{\ast }=0.5,1,2$ and 4. Radius shows (a) $A^{\ast }$, (b) $f^{\ast }$, (c) $C_{y}$ and (d) $\unicode[STIX]{x1D719}_{total}$. In (a–d) (●) $K_{p}^{\ast }=4$, (▪) $K_{p}^{\ast }=2$, (▴) $K_{p}^{\ast }=1$, (♦) $K_{p}^{\ast }=0.5$. (e,f) PSD contour plots of $f^{\ast }$ and $f_{C_{y}}^{\ast }$, respectively, for $K_{p}^{\ast }=4$. The spectral power is normalised by the maximum value at each $\unicode[STIX]{x1D719}_{rot}$ and is presented on a $\log _{10}$ scale. The black dashed line represents the natural response.

Figure 18.

Response of the sphere in the mode III transition regime ($U^{\ast }=14$) with imposed rotation over the range $0^{\circ }\leqslant$$\unicode[STIX]{x1D719}_{rot}\leqslant 360^{\circ }$ for four proportional gain values $K_{p}^{\ast }=0.5,1,2$ and 4. Radius shows (a) $A^{\ast }$, (b) $f^{\ast }$, (c) $C_{y}$ and (d) $\unicode[STIX]{x1D719}_{total}$. In (a–d) (●) $K_{p}^{\ast }=4$, (▪) $K_{p}^{\ast }=2$, (▴) $K_{p}^{\ast }=1$, (♦) $K_{p}^{\ast }=0.5$. (e,f) PSD contour plots of $f^{\ast }$ and $f_{C_{y}}^{\ast }$, respectively, for $K_{p}^{\ast }=2$. The spectral power is normalised by the maximum value at each $U^{\ast }$ and is presented on a $\log _{10}$ scale. The black dashed line represents the natural response.

3.4 Wake structures

As first illustrated by Govardhan & Williamson (Reference Govardhan and Williamson2005), and recently elucidated in great detail by Eshbal et al. (Reference Eshbal, Kovalev, Rinsky, Greenblatt and van Hout2019), the vibration of an elastically mounted sphere is primarily a result of the formation of two counter-rotating streamwise vortices. Several studies employing extensive PIV measurements of tethered spheres have further revealed the intricate aspects of fluid motion that lead to sustained vibration. In particular, the influence of the separating shear layer and weaker secondary vortices, in conjunction with sphere motion, on the overall wake structure has been detailed (van Hout, Krakovich & Gottlieb Reference van Hout, Krakovich and Gottlieb2010; Eshbal, Krakovich & van Hout Reference Eshbal, Krakovich and van Hout2012; Krakovich, Eshbal & van Hout Reference Krakovich, Eshbal and van Hout2013; van Hout, Katz & Greenblatt Reference van Hout, Katz and Greenblatt2013b). In the regime where large-amplitude vibration occurs, interaction between near wake vortices, the sphere, and the separating shear layer was found to lead to pinch-off of the dominant streamwise vortex structures. Krakovich et al. (Reference Krakovich, Eshbal and van Hout2013) estimated the fluid force acting on the sphere using the analogy between the two counter-rotating streamwise vortices seen here and wing-tip vortices in the wake of an aircraft. Comparable results were obtained, highlighting the importance of these dominant structures in sustaining vibration. Whilst evidently, the problem is strongly three-dimensional and dynamic in nature, here, phase-averaged PIV has been employed in the transverse ($y{-}z$) and equatorial ($x{-}y$) planes to reveal the effect of the imposed rotation on the timing of the dominant streamwise vortex structures that have been shown to be the primary contributor to sustained vibration. PIV in the transverse plane, located at $x/D=1.5$, reveals the development of streamwise vorticity over a vibration cycle. PIV in the equatorial plane offers further insight into how the imposed rotation alters the position of the wake structures in relation to the sphere.

When the imposed rotation nearly completely suppresses vibration, the wake resembles that of a fixed sphere, which over the Reynolds number range tested in this study ($3900\lesssim Re\lesssim 25\,800$), consists of vortices that shed at random azimuth angles each shedding cycle (Sakamoto & Haniu Reference Sakamoto and Haniu1990) (note that due to the presence of the mounting rod, there will be a bias to vortex shedding at particular azimuths as discussed in § 2.2). As a result, the phase-averaged PIV acquired in the $y{-}z$ plane in this study does not reveal the streamwise vortex structures for significantly suppressed vibration. Therefore, in this section, representative cases in the mode II and mode III transition regimes, where sufficient vibration exists to reveal the streamwise vortex structures, are examined to highlight changes in wake structure due to the imposed rotation.

Figure 19.

Streamwise vorticity shown in the cross-stream plane at $x/D=1.5$ in the mode II regime ($U^{\ast }=10$). (a,d,g) $K_{p}^{\ast }=0$. (b,e,h) $K_{p}^{\ast }=4$, $\unicode[STIX]{x1D719}_{rot}=0^{\circ }$. (c,f,i) $K_{p}^{\ast }=4$,$\unicode[STIX]{x1D719}_{rot}=180^{\circ }$. From top to bottom, each plot is separated by approximately a quarter period. The vertical black dashed lines show the extremities of sphere displacement. The black dashed circle shows the sphere location. The black arrows indicate the direction of motion. Blue and red contours show clockwise and anti-clockwise vorticity respectively. The normalised vorticity, $\unicode[STIX]{x1D714}^{\ast }=\unicode[STIX]{x1D714}D/U$, is depicted by the colour map over eight steps in the range $\unicode[STIX]{x1D714}^{\ast }\in [-2,2]$.

Figure 19 shows streamwise vorticity in the mode II regime with no imposed rotation, and two representative cases of imposed rotation at $\unicode[STIX]{x1D719}_{rot}=0^{\circ }$ and $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$, with $K_{p}^{\ast }=4$. The sphere location, which has been approximately adjusted allowing for the time for the streamwise vortex structures to convect to the imaging plane, is depicted by a dashed black circle outline. The extremities of sphere motion are depicted by vertical black dashed lines. For $K_{p}^{\ast }=4$ and $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$, vorticity has not been presented at the extremity of sphere displacement due to the minimal vorticity seen at these locations. Positions highlighting maximum vorticity were chosen instead. In these instances, the black arrows depict the direction of sphere motion. For the three cases illustrated in figure 19, it can be observed that with imposed rotation the wake still primarily consists of the counter-rotating vortex pair seen for the natural response. There are, however, changes to the spatial position and timing of these vortices.

It is difficult to make detailed observations about the strength and shape of vortex structures due to the phase-averaging process. Increasing vibration amplitude results in vorticity been averaged over a greater range of sphere displacement values. As a result of this averaging, with increased vibration, the bounds of the vortex pair stretches in the $y$-direction, causing the depiction of the vortex structure to morph from a circular to an elliptical shape as sphere motion increases and for the strength of the vortex structure to appear diminished.

We can observe, however, that imposed rotation with $K_{p}^{\ast }=4$ and $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$ results in a significant change in the timing of vortex formation, conforming with the results analysed in § 3.2, where it was reported that $\unicode[STIX]{x1D719}_{total}=10^{\circ }$. The wake structures observed for $K_{p}^{\ast }=4$ and $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$ resemble that presented by Govardhan & Williamson (Reference Govardhan and Williamson2005) for the mode I regime where $\unicode[STIX]{x1D719}_{total}\approx 40^{\circ }$.

For $K_{p}^{\ast }=4$ and $\unicode[STIX]{x1D719}_{rot}=0^{\circ }$, no appreciable change to the timing of the vortex structures is evident. It can be observed, however, that the imposed rotation alters the extent of the transverse motion of the vortex structures. For the natural response, the vortex structures predominately lie inside the extremities of sphere motion depicted by the vertical black dashed lines. For $K_{p}^{\ast }=4$ and $\unicode[STIX]{x1D719}_{rot}=0^{\circ }$, however, the vortex structures predominately lie outside the extremities.

Figure 20.

Streamwise vorticity shown in the cross-stream plane at $x/D=1.5$ past the peak of the mode II regime ($U^{\ast }=14$). (a,d,g) $K_{p}^{\ast }=0$. (b,e,h) $K_{p}^{\ast }=1$, $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$. (c,f,i) $K_{p}^{\ast }=4$, $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$. Refer to figure 19 for further details.

Due to the large displacements observed in the mode III transition regime, and limitations of the PIV set-up, figure 20 shows the streamwise vorticity component in the transverse plane over only half the vibration cycle in the mode III transition regime ($U^{\ast }=14$). For $K_{p}^{\ast }=0$, $K_{p}^{\ast }=1$ at $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$, and $K_{p}^{\ast }=4$ at $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$, as presented in § 3.2, the total phase was found to be $170^{\circ }$, $108^{\circ }$ and $49^{\circ }$, respectively. These changes in timing of the fluid force on the sphere are manifested in figure 20 as alterations to the timing of the streamwise vortex structures. From figure 20 it can be observed that increasing the proportional gain at $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$ results in increased vorticity as the sphere moves past $y/D=0$, and decreased vorticity at the extremities of sphere displacement.

Whilst it is beneficial to examine vorticity in the $y{-}z$ plane to observe the effect of rotation on the streamwise vortex structures, PIV in the equatorial plane highlights the deflection of the wake due to sphere vibration and imposed rotation. The effect of the imposed rotation is clearly visible in figure 21. It is evident that past the peak of mode II, where the rotation-induced vibration response is observed, the rotation alters the deflection of the wake. In comparison to the natural response, rotation at $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$ results in a wake deflection towards the trailing side of the sphere (in the transverse sense), whilst conversely, rotation at $\unicode[STIX]{x1D719}_{rot}=0^{\circ }$ results in a deflection towards the advancing side.

Figure 21.

Equatorial vorticity in the mode III transition regime ($U^{\ast }=14$). (a,d,g) $K_{p}^{\ast }=0$. (b,e,h) $K_{p}^{\ast }=4$, $\unicode[STIX]{x1D719}_{rot}=0^{\circ }$. (c,f,i) $K_{p}^{\ast }=4$, $\unicode[STIX]{x1D719}_{rot}=180^{\circ }$. From top to bottom, each plot is separated by a quarter period. The grey line shows the extremities of sphere displacement. The black circle shows sphere location. The black line depicts the approximate centre of the wake in the $x{-}y$ plane. Blue and red contours show clockwise and anti-clockwise vorticity respectively. The normalised vorticity, $\unicode[STIX]{x1D714}^{\ast }=\unicode[STIX]{x1D714}D/U$, is depicted by the colour map over eight steps in the range $\unicode[STIX]{x1D714}^{\ast }\in [-1,1]$.