4.1 FIV response of a tethered sphere

The flow past a tethered sphere of mass ratio $m^{\ast }=0.8$ with a tether length of $l^{\ast }=10$ was investigated by fixing the Reynolds number at a particular value. The sphere showed large-amplitude modes I and II vibration states at the larger Reynolds numbers investigated, similar to the observation of previous experimental studies. Figure 3 compares our results of the sphere vibration amplitude, $A_{z}^{\ast }=\sqrt{2}Z_{rms}/D$, at $Re=1200$ and $2000$ with the experimental results of Jauvtis et al. (Reference Jauvtis, Govardhan and Williamson2001) conducted with a sphere of the same mass ratio, where $Z$ is the sphere displacement in the $z$ direction. As can be seen, the present results match reasonably well with those of Jauvtis et al. Recall that, in the experiments, the Reynolds number was not fixed and varied approximately between $2000$ and $14\,000$. Consistent with their observations, the sphere response amplitudes at modes I and II were approximately $0.5D$ and $0.85D$, respectively.

Figure 3.

Comparison of the sphere response amplitude, $A_{z}^{\ast }$, at $Re=1200$ and $2000$ with the experimental results of Jauvtis et al. (Reference Jauvtis, Govardhan and Williamson2001) at higher and varying Reynolds numbers with a sphere of mass ratio $m^{\ast }=0.8$ over the reduced velocity range $U^{\ast }=[3,14]$. Consistent with their observations, the first two modes of sphere vibration states were observed.

In particular, at this mass ratio, Jauvtis et al. (Reference Jauvtis, Govardhan and Williamson2001) observed two peaks in the amplitude response curve, corresponding to modes I and II. In contrast, at $Re=1200$, the sphere response amplitude varied smoothly from mode I to mode II without a dip as seen in most of the amplitude response curves of tethered spheres (Jauvtis et al. Reference Jauvtis, Govardhan and Williamson2001; Govardhan & Williamson Reference Govardhan and Williamson2005). However, once the Reynolds number is increased to $2000$, the response curve formed a small peak for mode I as observed by Jauvtis et al. Therefore, if the Reynolds number is further increased, this local peak at mode I is expected to become more prominent. At $Re=1200$, the highest vibration amplitude occurred around $U^{\ast }=8.5$, and the vibration amplitude decreased beyond this point, in contrast to the observation of Jauvtis et al. (Reference Jauvtis, Govardhan and Williamson2001) of almost constant amplitude at larger $U^{\ast }$ values. However, as the Reynolds number is increased to $2000$, the response amplitudes were higher than at $Re=1200$. Moreover, the response curve was closer to their response curve. Again, if the Reynolds number is further increased, the response curve at mode II is also anticipated to become more similar to the response curve of Jauvtis et al. observed at higher and varying Reynolds numbers. These observations show that there is a considerable effect of Reynolds number on the sphere response in this Reynolds-number range. Rajamuni et al. (Reference Rajamuni, Thompson and Hourigan2018a) also showed that there is a substantial effect of the Reynolds number on FIV of an elastically mounted sphere with one degree of freedom (DOF) in the laminar regime.

Compared to the observation of Jauvtis et al. (Reference Jauvtis, Govardhan and Williamson2001), the predicted response curves look slightly shifted to the left. However, the response that was predicted at $U^{\ast }=4$ is not a periodic response as in mode I, albeit the shedding frequency is locked in to the oscillation frequency. Similar to the results of Jauvtis et al., we also expected mode I to occur around $U^{\ast }=5$ as the static vortex shedding frequency is $f_{vor}\approx 0.2$. Therefore, at $U^{\ast }=4$, the sphere may be in transition from no oscillation to VIV. Nonetheless, we observed mode I vibration at a slightly lower reduced velocity compared to that observed by Jauvtis et al. (Reference Jauvtis, Govardhan and Williamson2001). This difference may be due to the effect of Reynolds number. Note that, for the predictions, the Reynolds number was fixed, whereas it was allowed to vary with $U^{\ast }$ in the experiments.

Figure 4.

The sphere response curves at $Re=500$ and $1200$ of a tethered sphere of $m^{\ast }=0.8$: the normalized amplitude, $A_{z}^{\ast }$, plotted against (a) the reduced velocity, $U^{\ast }$, and (b) the normalized velocity, $(U^{\ast }/f^{\ast })St$.

To explore the effect of Reynolds number in the laminar regime, another set of simulations was conducted at $Re=500$ with the same mass ratio and tether length. Figure 4(a) shows a comparison of the sphere response amplitude of $Re=500$, $1200$ and $2000$ for the reduced velocity range $U^{\ast }=[3,32]$. As can be seen, the sphere response amplitude increased globally as the Reynolds number increased over $U^{\ast }\in [4.5,16]$. As discussed earlier, it showed periodic mode I and II vibrations over $U^{\ast }\approx [4.5,7]$ and $[8,16]$, respectively, at $Re=1200$ and $2000$. At $Re=500$, a periodic vibration response was found over $U^{\ast }=[4.5,12]$. The response curve took a bell shape with a maximum amplitude of ${\approx}0.45D$. At this Reynolds number, modes I and II were not able to be distinguished clearly from the amplitude response curve alone. Govardhan & Williamson (Reference Govardhan and Williamson2005) found that mode I occurred around $(U^{\ast }/f^{\ast })St=1$ while mode II occurred in the range of $(U^{\ast }/f^{\ast })St\approx [1.4,2]$. However, if the amplitude is plotted against $(U^{\ast }/f^{\ast })St$, both modes I and II responses became clear, even at $Re=500$, from the range of $(U^{\ast }/f^{\ast })St$ (see figure 4b).

Similar to the $Re=500$ case, Rajamuni et al. (Reference Rajamuni, Thompson and Hourigan2018a) also observed bell-shaped response curves at $Re=300$ and $800$ with an elastically mounted sphere; these were named ‘branch A’. Moreover, the maximum amplitudes they observed at those Reynolds numbers were ${\approx}0.4D$ and $0.6D$, respectively. This observation leads to the conclusion that a sphere displays a trend of increasing amplitude globally over $U^{\ast }\approx [4,16]$, as the Reynolds number increased from $300$ to $2000$, regardless of whether it was an elastically mounted or a tethered sphere.

For $U^{\ast }>16$, the sphere showed an aperiodic response for all three Reynolds numbers. Although the amplitudes were scattered, they showed initially an increasing trend and then levelling off around $0.8D$ as the reduced velocity increased to $34$. Rajamuni et al. (Reference Rajamuni, Thompson and Hourigan2018a) also reported a similar behaviour at $Re=800$ for $U^{\ast }>13$, but with a purely increasing trend up to $U^{\ast }=50$. As Rajamuni et al. claimed, this intermittent burst of vibration strongly resembles mode IV vibration discovered by Jauvtis et al. (Reference Jauvtis, Govardhan and Williamson2001) with a heavy sphere of $m^{\ast }=80$ for $U^{\ast }>100$. This begs the question: ‘Why is mode IV observed just after mode II without an intervening mode III?’ A discussion of this is given in § 4.3.

Figure 5.

The FIV response of a tethered sphere at $Re=1200$ over the reduced velocity range $U^{\ast }=[3,32]$: (a) sphere vibration amplitudes $A_{x}^{\ast }$, $A_{y}^{\ast }$ and $A_{z}^{\ast }$ in the streamwise, lateral and transverse directions, respectively; (b) the periodicity of the sphere vibration, $\unicode[STIX]{x1D706}_{A}=\sqrt{2}Z_{rms}/Z_{max}$; and (c) frequency ratio, $f^{\ast }=f/f_{n}$.

Williamson & Govardhan (Reference Williamson and Govardhan1997) and Govardhan & Williamson (Reference Govardhan and Williamson2005) identified that the motion of a tethered sphere is principally in the transverse direction. Consistently, we also observed large-amplitude vibrations in the transverse ($z$) direction compared to the streamwise ($x$) and lateral ($y$) directions (see figures 5a and 6a). For mode I and II regimes, the sphere showed a negligible amplitude in the lateral direction while displaying a small amplitude of ${\approx}0.08D$ in the streamwise direction, as shown in figure 5(a) at $Re=1200$. This is consistent with the observation of Williamson & Govardhan (Reference Williamson and Govardhan1997) with a sphere of $m^{\ast }=0.73$ and tether length ratios $l^{\ast }=8.9$ and $3.8$.

The periodicity of the sphere vibration, $\unicode[STIX]{x1D706}_{A}$, is defined as $\sqrt{2}Z_{rms}/Z_{max}$, where $Z_{max}$ is the maximum amplitude observed at each $U^{\ast }$. According to this definition, $\unicode[STIX]{x1D706}_{A}$ take values from $0$ to $1$, with $\unicode[STIX]{x1D706}_{A}=1$ representing the most periodic response. For both modes I and II, the sphere vibrations were highly periodic. However, the vibration was more periodic at the peak of mode II compared with the response at mode I (see figures 5b and 7a,c). For both modes I and II, the sphere vibrated in synchrony with the vortex shedding frequency, $f_{s}$, and was close to the system’s natural frequency, as expected (see figure 5c). Govardhan & Williamson (Reference Govardhan and Williamson1997, Reference Govardhan and Williamson2005) observed for light spheres ($m^{\ast }<1$) across the mode II regime and above that the dynamic vortex shedding frequency remained between the static body vortex shedding frequency, $f_{vor}$, and the natural frequency of the system.

Figure 6.

The FIV response of a tethered sphere at $Re=500$ over the reduced velocity range $U^{\ast }=[3,32]$: (a) sphere vibration amplitude, $A^{\ast }$; (b) the periodicity of the sphere vibration, $\unicode[STIX]{x1D706}_{A}=\sqrt{2}Z_{rms}/Z_{max}$; and (c) frequency ratio, $f^{\ast }=f/f_{n}$.

As the sphere response curve for $Re=500$ deviated from the response curve observed at higher Reynolds numbers over the mode I and II regimes ($U^{\ast }=[4.5,12]$), a negligible amplitude was observed in both the streamwise ($x$) and lateral ($y$) directions (see figure 6a). Vibrations in the transverse ($z$) directions were highly periodic as in the peak of mode II (figure 6b). This is indeed a VIV response ($f=f_{s}=f_{n}$). From these observations, we can predict that the response of a tethered sphere collapses well with the response of an elastically mounted sphere of one DOF for low Reynolds numbers over the modes I and II regimes.

As the reduced velocity increased beyond $U^{\ast }=12$ (beyond the mode II regime), the periodicity of the sphere response gradually decreased at $Re=1200$ and reached a value of ${\approx}0.5$ for the mode IV regime (see figure 5b). A similar behaviour was observed at $Re=500$ as well. However, as $U^{\ast }$ was increased, mode IV appeared quickly after the periodic vibrations and the periodicity of the response was even lower. The sphere vibration was intermittent in mode IV, as shown in figure 7(e) at $U^{\ast }=27.4$, and it followed an irregular trajectory without a clear pattern, as shown in figure 7(f). As $\unicode[STIX]{x1D706}_{A}$ decreased, the sphere displacement in the streamwise and lateral directions became slightly more significant. The streamwise amplitude was almost a constant value of ${\approx}0.3D$ at $Re=1200$ and ${\approx}0.2D$ at $Re=500$ over mode IV. The lateral amplitude showed a linearly increasing trend as the reduced velocity increased beyond $16$ and was almost half the transverse amplitude at the highest reduced velocity considered ($U^{\ast }=32$) at both Reynolds numbers. This shows that the randomness of the signal increases with reduced velocity.

Figure 7.

(a,c,e) Time histories and (b,d,f) sphere trajectories in the $x$–$z$ plane at $Re=1200$, where $\unicode[STIX]{x1D70F}=tU/D$ is the non-dimensional time and $T$ is the period: (a,b) at $U^{\ast }=6$ (mode I), (c,d) at $U^{\ast }=9$ (mode II), and (e,f) at $U^{\ast }=27.4$ (mode IV).

Figure 8 displays the time-averaged drag coefficient, $\overline{C}_{d}$, as a function of the reduced velocity for all three Reynolds numbers considered. An increment in $\overline{C}_{d}$ is observed when it vibrated periodically, as found in previous experimental and numerical studies (Gottlieb Reference Gottlieb1997; Williamson & Govardhan Reference Williamson and Govardhan1997; Behara et al. Reference Behara, Borazjani and Sotiropoulos2011). At $Re=500$, as the sphere began to vibrate at $U^{\ast }=4.5$, an ${\sim}33\,\%$ increment of $\overline{C}_{d}$ was observed from the value for a stationary sphere. Similar to the observation of Rajamuni et al. (Reference Rajamuni, Thompson and Hourigan2018a) for their branch A, this increment then decreased over the synchronization regime. However, they observed a sharp turn at the beginning of the range, while we observed a smooth turn. At both $Re=1200$ and $2000$, $\overline{C}_{d}$ showed another jump at the beginning of mode II, as Govardhan & Williamson (Reference Govardhan and Williamson1997) observed. The increments of $\overline{C}_{d}$ were ${\sim}60\,\%$ and ${\sim}100\,\%$ at modes I and II, respectively. In the mode IV regime, even larger-amplitude vibrations are observed, whereas the drag coefficient hardly changed at all. This is because mode IV is not a VIV response, as discussed in § 4.4.

Figure 8.

Variation of the time-averaged drag coefficient, $\overline{C}_{d}$, with the reduced velocity at $Re=500$, $1200$ and $2000$. The horizontal grey lines show $\overline{C}_{d}$ calculated with a stationary sphere at $Re=500$ (dotted line) and $Re=1200$ (solid line).

Figure 9.

The time-mean layover angle, $\bar{\unicode[STIX]{x1D703}}_{L}$ (angle of tether from the $y$ direction), as a function of reduced velocity. The dotted lines represent the estimated $\bar{\unicode[STIX]{x1D703}}_{L}$ from the drag coefficient at each Reynolds number.

The time-mean layover angle, $\overline{\unicode[STIX]{x1D703}}_{L}$, is defined as the angle of the tether to the lateral direction (vertical). An exponentially increasing trend of $\overline{\unicode[STIX]{x1D703}}_{L}$ with a slight variation in mode II was observed for all of the Reynolds numbers considered (see figure 9). Since the lift force is comparatively small, the layover angle can be estimated from the non-dimensional drag and buoyancy forces as

(4.1)$$\begin{eqnarray}\displaystyle \tan (\bar{\unicode[STIX]{x1D703}}_{L})=\frac{C_{d}}{(1-m^{\ast })\unicode[STIX]{x1D6FC}}. & & \displaystyle\end{eqnarray}$$

As can be seen in figure 9, the calculated $\overline{\unicode[STIX]{x1D703}}_{L}$ is coincident with the estimated values. Note that, even though $\overline{C}_{d}$ is constant over the mode IV regime, $\tan (\unicode[STIX]{x1D703}_{L})$ increases; this is because $\unicode[STIX]{x1D6FC}=4/(3Fr^{2})$ is not constant in this numerical study.

In this study, the tether was assumed rigid and the torsion of the sphere was neglected. However, it is worth exploring the nature of the sphere response when the sphere is allowed to rotate through the axis of the tether, according to the moment acting on it. Nevertheless, the moments acting on the sphere were found to be negligible compared to the forces acting on it (moments ${<}1\,\%$ forces). Thus, it can be assumed that the sphere undergoes a very small amount of rotation when it is also subjected to torsion. Our previous study (Rajamuni et al. Reference Rajamuni, Thompson and Hourigan2018b) and the study of Sareen et al. (Reference Sareen, Zhao, Lo Jacono, Sheridan, Hourigan and Thompson2018a) investigated the VIV response of an elastically mounted sphere under a forced rotation. Both of these studies found that the effect of rotation is negligible on the VIV response of the sphere, for small rotation rates. The investigation of Sareen et al. (Reference Sareen, Zhao, Sheridan, Hourigan and Thompson2018c) also found that the rotary motion of the sphere has a negligible effect on the VIV response of a sphere, for small velocity ratios. This combined evidence indicates that the VIV response of a tethered sphere is highly unlikely to be altered by the sphere torsion.

From the findings of this study and previous experimental and numerical studies, a brief discussion of the different modes of sphere vibrations is provided in the following three subsections.

4.2 Modes I and II

Modes I and II are the only VIV responses observed for a sphere out of the four vibration modes found. Mode I response is due to a natural resonance, where the vibration of the sphere is excited when the natural frequency of the system, $f_{n}$, is close to the stationary-body (non-VIV) vortex shedding frequency. For example, at $Re=1200$, the vortex shedding frequency of a stationary sphere, $f_{vor}$, is approximately $0.2$, which means that the natural oscillation frequency and the stationary-body vortex shedding frequency coincide at $U^{\ast }=5$. As a result of this, mode I was observed close to $U^{\ast }=5$ (see figure 5a).

In almost all of the VIV studies, the sphere showed large-amplitude periodic vibrations even after the resonance range, and this is known as the mode II vibration state. As the sphere is allowed to translate freely, a reduction is observed in the vortex shedding frequency from the value for a static sphere. In this manner, the dynamic shedding frequency deviated from the static shedding frequency and synchronized with the natural frequency (see figures 5c and 6c). As a result, the sphere showed large-amplitude vibration (mode II) after the resonance state.

The mode II response was observed beyond the mode I regime as the reduced velocity was increased. The sphere response amplitude varied smoothly as the vibration state transitioned from mode I to mode II, in contrast to the distinct branches observed in the amplitude response curve of an elastically mounted cylinder (one DOF). The sphere response curve showed a small local peak in mode I for light tethered spheres ($m^{\ast }<1$); for example, the observations of Williamson & Govardhan (Reference Williamson and Govardhan1997) at $m^{\ast }=0.729$ with both $l^{\ast }=3.83$ and $8.93$ and at $m^{\ast }=0.082$ with $l^{\ast }=9.28$, those of Govardhan & Williamson (Reference Govardhan and Williamson1997) at $m^{\ast }=0.26$, and those of Jauvtis et al. (Reference Jauvtis, Govardhan and Williamson2001) at $m^{\ast }=0.8$, can be mentioned. However, the local peak in mode I is obscured for heavy spheres ($m^{\ast }>1$), especially for elastically mounted spheres, as discussed by Govardhan & Williamson (Reference Govardhan and Williamson2005). The range of $U^{\ast }$ values varies for different modes; mode I is observed for a very short range (within ${\sim}(1{-}2)U^{\ast }$) compared with mode II. This is expected, as mode I is the result of resonance.

The difference between modes I and II was studied by Govardhan & Williamson (Reference Govardhan and Williamson2005) by analysing the phase between sphere oscillation and the fluid forces acting on the sphere. Lighthill (Reference Lighthill1986) showed that the total fluid force, $F_{t}$, can conveniently be split into two components, a ‘potential force’ component ($F_{p}$) related to the potential added mass and a ‘vortex force’ component ($F_{v}$) related to the dynamic vorticity. For a cylinder, Govardhan & Williamson (Reference Govardhan and Williamson2000) found a shift in the total phase, $\unicode[STIX]{x1D719}_{t}$ (the phase between the sphere displacement and the total force), or the vortex phase, $\unicode[STIX]{x1D719}_{v}$ (the phase between the sphere displacement and the vortex force), as the vibration state transitions from one branch to another. Analogously, Govardhan & Williamson (Reference Govardhan and Williamson2005) observed a shift in the vortex phase, while the total phase remains almost constant, as the sphere transitions from mode I to mode II. In a similar fashion, we found that the vortex phase shifted from $0^{\circ }$ to $180^{\circ }$ as the vibration state transitioned from mode I to mode II, while the total phase remained at ${\approx}0^{\circ }$, as shown in figure 10(c,d) for $Re=500$ and $1200$, respectively.

For an elastically mounted cylinder, Govardhan & Williamson (Reference Govardhan and Williamson2000) argued that a sudden shift in total or vortex phase can be expected as the body oscillation frequency passes the natural frequency of the system (or the frequency ratio crosses the $f^{\ast }=1$ line). Consistent with this argument, we observed that the shift in $\unicode[STIX]{x1D719}_{v}$ occurs as the frequency ratio crosses the $f^{\ast }=1$ line (see figure 10e,f). In addition to this, Govardhan & Williamson discussed that, for a purely sinusoidal response with zero damping ratio, the total or vortex force can only be either phase-aligned or be $180^{\circ }$ out of phase with the body vibration. That discussion related to an elastically mounted cylinder. The motion of a tethered sphere has no damping effect (see equation (2.21)). Therefore, if the sphere response is periodic, we can predict that $\unicode[STIX]{x1D719}_{t}$ and $\unicode[STIX]{x1D719}_{v}$ can be either $0^{\circ }$ or $180^{\circ }$ in a similar way. Consistently, we observed that $\unicode[STIX]{x1D719}_{t}$ and $\unicode[STIX]{x1D719}_{v}$ were mostly either $0^{\circ }$ or $180^{\circ }$.

Figure 10.

Variation of (a,b) sphere response amplitude, and (c,d) the total phase, $\unicode[STIX]{x1D719}_{t}$, and vortex phase, $\unicode[STIX]{x1D719}_{v}$, over modes I and II regimes. (e,f) The frequency ratio crosses the $f^{\ast }=1$ line as the vortex phase shifts from $0^{\circ }$ to $180^{\circ }$. Column (a,c,e) for $Re=500$ and (b,d,f) for $Re=1200$.

The response amplitude of mode II was found to be higher than the amplitude of mode I (for some cases, approximately twice the amplitude of mode I) by experimental studies at higher Reynolds numbers. We also observed a similar behaviour at $Re=1200$ and $2000$. However, for $Re<1000$, the amplitude of mode II was observed to be similar to that of mode I (present study at $Re=500$ and Rajamuni et al. (Reference Rajamuni, Thompson and Hourigan2018a) at $Re=300$ and $800$). Therefore, the effect of Reynolds number on the amplitude of mode II is not negligible in the laminar regime.

4.2.1 Effect of Reynolds number

Govardhan & Williamson (Reference Govardhan and Williamson2005) showed that the effect of Reynolds number is negligible for the VIV of a sphere over a range of $Re\in [2000,12\,000]$ with the help of the Griffin plot and the tether length ratio. However, the effect of Reynolds number is significant over $Re\in [300,2000]$. Figure 11 plots the sphere predicted responses together with the results of Rajamuni et al. (Reference Rajamuni, Thompson and Hourigan2018a) with elastically mounted spheres at five different Reynolds numbers over modes I and II regimes. As can be seen, the response amplitude increases globally with the Reynolds number. This effect is more significant over the mode II regime than the mode I regime. In addition, the mode II regime widened as the Reynolds number increased in this range. Mode II appears to be more sensitive to the Reynolds number.

Figure 11.

The amplitude response curves at different Reynolds numbers at the modes I and II regimes: ▹, at $Re=300$ of Rajamuni et al. (Reference Rajamuni, Thompson and Hourigan2018a) with an elastically mounted sphere of $m^{\ast }=2.865$; ▫, at $Re=500$; ♢, at $Re=800$ of Rajamuni et al. (Reference Rajamuni, Thompson and Hourigan2018a) with an elastically mounted sphere of $m^{\ast }=2.865$; ●, at $Re=1200$; and $\boldsymbol{\times }$ at $Re=2000$.

Figure 12.

Wake structures in modes I and II at $Re=500$, $1200$ and $2000$ visualized with isosurface of $Q$ at $0.01$. Flow from left to right.

Figure 13.

Results of DMD analysis of the 2-D velocity field over $10$ oscillation cycles of mode I at $Re=500$: (a) plot of eigenvalues of the companion matrix; (b) frequency spectrum; (c–f) dynamic modes KM (visualized by vorticity field) corresponding to frequencies $f_{0}$, $f_{1}$, $2f_{1}$ and $3f_{1}$, respectively; (g) actual vorticity field; (h–j) reconstructions of the vorticity field using all the KMs, only the dominant KM ($f_{0}+f_{1}$), and the dominant KM and its higher-order harmonics ($f_{0}+f_{1}+2f_{1}+3f_{1}+4f_{1}+5f_{1}+6f_{1}$), respectively.

4.2.4 Dynamic mode decomposition

For further examination of the wake, and to identify the dominant wake modes, DMD was performed. The nature of the wake of mode I ($U^{\ast }=5.5$) was studied at $Re=500$ using the two-dimensional (2-D) velocity field (on the $x$–$z$ plane) over 10 oscillation cycles with 23 snapshots per cycle. As figure 13(a) shows, the plot of eigenvalues of the companion matrix lies in a unit circle, indicating a periodic wake. Moreover, the frequency, $f_{1}$, of the dynamic mode with the highest magnitude was identical to the oscillation frequency of the sphere, as expected. The sphere oscillation was not purely sinusoidal in this case; therefore, the frequency spectrum contained other frequencies besides the dominant frequency and its higher-order harmonics (see figure 13b). The vorticity field was used to visualize the Koopman modes (KMs), and to compare the reconstruction using these modes with the actual field (see figure 13c–j). The reconstruction with all modes, shown in figure 13(h), is identical to the actual field shown in figure 13(g), providing some validation to the analysis. The dominant KM has captured the main features of the field (see figure 13i). Moreover, when its higher-order harmonics are also used, it looks much closer to the actual field – compare figure 13(j) and (g).

The DMD analysis of the 2-D velocity field of mode II ($U^{\ast }=9$) at $Re=1200$ over $24$ oscillation cycles with $48$ snapshots per cycle was performed, and results are presented in figure 14. The plot of eigenvalues of the companion matrix provides evidence of a highly periodic wake. The sphere oscillation is purely sinusoidal in mode II. Inferred from this, the frequency spectrum in mode II for $Re=500$ (not shown) contained only the dominant frequency and its higher-order harmonics. However, as $Re$ was increased to $1200$, the frequency spectrum was dense with other frequencies, showing the effect of Reynolds number (see figure 14b). Nevertheless, the underlying streamwise vortex structure observed for the dominant KM (see figure 14e) was synchronized with the sphere oscillation, so the sphere showed large-amplitude mode II vibrations. This was also clearly visible from the isosurfaces of $Q$ when 3-D velocity fields were used for the analysis, as shown in figure 15 for both modes I and II.

Figure 14.

Results of DMD analysis of the 2-D velocity field over $24$ oscillation cycles of mode II at $Re=1200$: (a) plot of eigenvalues of the companion matrix; (b) frequency spectrum; (c) actual vorticity field; (d–f) reconstructions of the vorticity field using all the KMs, only the dominant KM ($f_{0}+f_{1}$), and the dominant KM and its higher-order harmonics ($f_{0}+f_{1}+2f_{1}+3f_{1}+4f_{1}$), respectively.

Figure 15.

Isosurfaces of the wake at modes I and II reconstructed from all of the KMs (a,b) and with only the dominant KM (c,d). The dominant KM has captured the main feature of the flow.

4.3 Mode III

After the mode II regime, Jauvtis et al. (Reference Jauvtis, Govardhan and Williamson2001) discovered another periodic vibration state, namely mode III. It was an unexpected finding that was first observed with a sphere of $m^{\ast }=28$ with water channel experiments. To study this mode further, they performed a set of wind tunnel experiments with a tethered sphere of $m^{\ast }=80$ for a wide range of $U^{\ast }$. Mode III was found to occur after a desynchronization regime for a broad range of $U^{\ast }$ from $20$ to $40$. They repeated this set of experiments with a larger tunnel to check whether it was an experimental artifact. Mode III was evident even with the larger tunnel. In addition to that, it was observed with a sphere of $m^{\ast }=940$. The sphere response was remarkably periodic and the vibration amplitudes were almost the same as for mode II. Not only tethered spheres, but also elastically mounted spheres, showed mode III vibrations, but without a desynchronization regime; for example, this was found by Govardhan & Williamson (Reference Govardhan and Williamson2005) with $m^{\ast }=53.7$ and Sareen et al. (Reference Sareen, Zhao, Lo Jacono, Sheridan, Hourigan and Thompson2018a) with $m^{\ast }=14.2$.

As a consequence of the high mass ratio in previous experiments, the oscillation frequency of mode III was identical to the system’s natural frequency. Nevertheless, it was difficult to explain the existence of mode III, since the principal vortex shedding frequency is 3–8 times higher than the oscillation frequency. Govardhan & Williamson (Reference Govardhan and Williamson2005) observed multiple small-scale structures in the wake of mode III. There was no clear association between vortex shedding and full or half wavelength of the sphere vibration.

Mode III is not possible to explain with the classic lock-in theories and such a vibration state does not exist for the case of cylinder free vibration. Govardhan & Williamson (Reference Govardhan and Williamson2005) argued that the flow must create a forcing on the body at this low frequency, sufficient to deliver a net energy transfer to the body motion. They measured the streamwise vorticity using DPIV and found a two-sided chain of trailing vortex pairs that is locked in with the sphere frequency. With this observation, they claimed that there is a net positive energy transfer in the vibration over a cycle, enabling the highly periodic mode III. A tethered or an elastically mounted sphere is intrinsically unstable. Govardhan & Williamson (Reference Govardhan and Williamson2005) argued that, if the sphere is perturbed in the transverse direction, it can generate a self-sustaining vortex force to enhance the body vibration, to possibly a large amplitude. In the mode III regime, the sphere is highly likely to be perturbed, as its wake is naturally responsive to low-frequency disturbances (Brücker Reference Brücker2001). Hence, Govardhan & Williamson (Reference Govardhan and Williamson2005) concluded that mode III is an example of ‘movement-induced vibration’, categorized by Naudascher & Rockwell (Reference Naudascher and Rockwell2012).

Compared to the first two vibration modes, mode III has been little examined; only a couple of studies have reported it, and its nature is not well understood yet. Therefore, further investigation is presented here. We attempt to enhance the understanding of mode III, through some previous experimental observations together with selected simulations.

We observed that the mode III state appears only with heavy spheres. In particular, Jauvtis et al. (Reference Jauvtis, Govardhan and Williamson2001) found mode III for spheres of $m^{\ast }=28$, $80$ and $940$, while Govardhan & Williamson (Reference Govardhan and Williamson2005) observed it for spheres of $m^{\ast }=11.7$, $31.1$, $53.7$ and $75$, and Sareen et al. (Reference Sareen, Zhao, Lo Jacono, Sheridan, Hourigan and Thompson2018a) for a sphere of $m^{\ast }=14.2$. From this, we can hypothesize that mode III arises only for high-inertia systems (heavy spheres). We intended to check this hypothesis with a sphere of higher mass ratio. Unfortunately, simulations are very costly for higher-mass-ratio cases, since it takes a very long time to reach the steady state. Therefore, only a couple of runs were feasible.

At $m^{\ast }=0.8$ and $Re=1200$, intermittent mode IV vibrations were observed for $U^{\ast }\in [16,32]$ just after mode II, without a trace of mode III. To study the possibility of mode III being excited, we performed a simulation with a sphere of $m^{\ast }=80$ at $U^{\ast }=30$ and $Re=1200$. This $U^{\ast }$ was chosen since mode III emerged predominantly for the reduced velocity range $U^{\ast }\in [20,40]$ in earlier experimental studies. Figure 16(a) shows the time history of the sphere displacement in this case. As can be seen, the sphere vibration has become fairly periodic from intermittent vibration as the mass ratio increased to $80$. The sphere response amplitude converged to a value of ${\approx}0.5D$ after a long transient period. The sphere vibration is highly periodic. Moreover, its frequency coincides with the system’s natural frequency, as expected at this high mass ratio. However, neither the total nor vortex forces were periodic, and those forces were small in magnitude with a high frequency (see figure 16b). As seen in previous experimental studies, the vortex shedding frequency was higher than the sphere frequency; for this case, it was approximately six times higher. Thus, this is essentially a mode III response. Figure 17 shows the wake structure observed. Figure 16(c,d) shows the sphere trajectory in $y$–$z$ and $x$–$z$ planes. Its displacement in the streamwise ($x$) and lateral ($y$) directions were negligible compared to the displacement in the transverse ($z$) direction. However, the streamwise frequency is the same as the transverse frequency, in contrast to modes I and II, for which the streamwise frequency is double the transverse frequency. Therefore, the sphere trajectory in mode III deviated from the figure ‘8’ shape to a figure ‘0’ shape. This indicates that mode III is essentially not an extension of mode II, but a completely different vibration state.

Figure 16.

The sphere response at $m^{\ast }=80$, $U^{\ast }=30$ and $Re=1200$: (a) time history; (b) sphere displacement together with total and vortex force coefficients, $C_{t}$ and $C_{v}$, respectively; and (c,d) trajectory of the sphere in $y$–$z$ and $x$–$z$ planes, respectively, at the steady state.

Figure 17.

Wake observed in mode III ($Re=1200$, $m^{\ast }=80$, $l^{\ast }=10$ and $U^{\ast }=30$) visualized with isosurface of $Q=0.001$. Flow from left to right.

Figure 18.

The DMD analysis of mode III ($Re=1200$, $m^{\ast }=80$) for $10$ oscillation cycles with $48$ snapshots per cycle: (a) plot of eigenvalues of the companion matrix; (b) frequency spectrum; and (c,d) isosurfaces of $Q$ using all KMs and only the dominant KM conjugate pair, respectively.

4.3.3 Effect of the mass ratio

Figure 19.

The sphere response at $m^{\ast }=80$ and $Re=1200$. (a) Comparison of the response amplitudes for $m^{\ast }=80$ calculated for $U^{\ast }=30$, $70$ and $150$ denoted by $\odot$ with the amplitude of $m^{\ast }=0.8$ denoted by ●. The dotted line shows the expected response amplitude curve for $m^{\ast }=80$. (b,c,d) The time histories at $m^{\ast }=80$ for $U^{\ast }=30$, $70$ and $150$, respectively.

Figure 20.

The sphere response as a function of mass ratio at $Re=1200$ and $U^{\ast }=30$: (a) the amplitude response in all three directions and (b) periodicity of the sphere response.

Figure 21.

Time history of the sphere displacement in the $z$ direction for $30$ oscillation cycles for the mass ratios, $m^{\ast }=0.8$, $3$, $10$, $20$, $30$, $40$, $60$ and $80$. The Reynolds number of the flow is $Re=1200$ and the reduced velocity is ${\approx}30$. The sphere vibration smoothly transitioned from aperiodic to periodic as the mass ratio increased.

The self-excitation initiated by the sphere wake pattern in this low-frequency regime becomes regular most likely as a result of the high inertia of the system. If the mass ratio is too small, then the sphere motion may not become regular and will show random vibration (mode IV), as observed at $m^{\ast }=0.8$. To investigate this issue further, a set of simulations was conduced at $Re=1200$ and $U^{\ast }\approx 30$ by varying the mass ratio from $0.8$ to $80$. The sphere was found to vibrate in an irregular manner (mode IV) at $m^{\ast }=0.8$. Although the sphere mainly vibrated in the transverse ($z$) direction, the components in the streamwise and the lateral directions were significant. This response will be discussed in detail in the next subsection. As figure 20(a) shows, the response amplitudes in the streamwise and lateral directions decayed with increasing mass ratio. Figure 21 shows the time history of the sphere vibration in the $z$ direction for the mass ratios $m^{\ast }=0.8$, $3$, $10$, $20$, $40$, $60$ and $80$. The irregular sphere response observed at $m^{\ast }=0.8$ gradually regulated as the mass ratio increased. The periodicity of the signal became higher as the mass ratio increased (see figure 20b). The sphere appears to show mode III response beyond $m^{\ast }\approx 20$.

Figures 22 and 23 show the phase plots and the Poincaré maps generated from the transverse displacement of the sphere at the above-mentioned mass ratios. The phase plot is somewhat disordered for low values of $m^{\ast }$, while the phase plot is regular, showing a circular orbit, at higher values of $m^{\ast }$. Consistently, the Poincaré maps are spread out for small mass ratios, and for higher mass ratios the maps are concentrated at the origin. These observations provide evidence of a chaotic response (mode IV) for $m^{\ast }=0.8$, $3$ and $10$, and a regular response for $m^{\ast }\geqslant 20$. As can be seen from figures 22 and 23, the transition from mode IV (chaotic) to mode III (regular) with increasing mass ratios is a smooth transition.

4.3.4 Robustness of mode III

The mode III state can be identified as an unstable state that can only appear for high-inertia spheres. Moreover, mode III appears quite sensitive to disturbances. For example, Sareen et al. (Reference Sareen, Zhao, Lo Jacono, Sheridan, Hourigan and Thompson2018a) observed a mode III type of response for $U^{\ast }>14$ for an elastically mounted sphere with zero rotation. However, when a rotation was imposed on the sphere, mode III was no longer observed. Mode III seems weaker than mode II. As a result, we can conclude that mode III is likely to disappear if a continual disturbance is applied on the system.

Figure 22.

Phase plots of transverse displacement, $Z$, for $m^{\ast }=0.8$, $3$, $10$, $20$, $30$, $40$, $60$ and $80$, at $Re=1200$ and $U^{\ast }\approx 30$.

Figure 23.

Poincaré maps for $Z/D$, taken from each upward zero crossing of the sphere transverse displacement for various mass-ratio values: $m^{\ast }=0.8$, $3$, $10$, $20$, $30$, $40$, $60$ and $80$, at $Re=1200$ and $U^{\ast }\approx 30$.

4.4 Mode IV

Jauvtis et al. (Reference Jauvtis, Govardhan and Williamson2001) observed intermittent bursts of large-amplitude vibration (mode IV) after the mode III regime with a sphere of $m^{\ast }=80$ for $U^{\ast }>100$. The periodicity of mode IV was found to be $\unicode[STIX]{x1D706}_{A}=0.5$ compared to the highly periodic first three modes ($\unicode[STIX]{x1D706}_{A}=1$). They found that the sphere vibration frequency remained very close to the system’s natural frequency throughout the range of velocity up to at least $U^{\ast }=300$. Jauvtis et al. argued that mode IV cannot be a VIV response as the vortex shedding frequency is much higher than the sphere frequency and there is no correlation between those two frequencies.

Rajamuni et al. (Reference Rajamuni, Thompson and Hourigan2018a) also observed a mode IV type of aperiodic vibration in their numerical study at $Re=800$, and called it the intermittent branch. It was observed immediately following their periodic branch A for $U^{\ast }>14$ up to $U^{\ast }=50$ with a sphere of $m^{\ast }=2.685$. In mode IV, Rajamuni et al. (Reference Rajamuni, Thompson and Hourigan2018a) found that the r.m.s. of the oscillation amplitude linearly increased with increasing $U^{\ast }$, similar to the observation of Jauvtis et al. (Reference Jauvtis, Govardhan and Williamson2001). It was quite surprising how a small-mass-ratio sphere could show mode IV response for relatively low reduced velocities. Rajamuni et al. (Reference Rajamuni, Thompson and Hourigan2018a) conjectured that it may be an effect of zero structural damping. Moreover, they argued that an increased damping may reduce or even suppress these intermittent vibrations.

Figure 24.

Sphere trajectories in mode IV and $Re=1200$ ($m^{\ast }=0.8$). Top row in $y$–$z$ plane and bottom row in $x$–$z$ plane at $U^{\ast }=13.9$, $17.7$, $22.2$, $27.3$ and $32$.

Mode IV type intermittent response was observed in this numerical study with a tethered sphere of $m^{\ast }=0.8$ at all three Reynolds numbers considered. In particular, the sphere showed mode IV at $Re=500$ for $U^{\ast }>14$ and at $Re=1200$ for $U^{\ast }>16$. For this mass ratio, mode IV appeared immediately after mode II without an intervening mode III, as Rajamuni et al. (Reference Rajamuni, Thompson and Hourigan2018a) observed with an elastically mounted sphere. Figure 24 shows the trajectories of the sphere at five different $U^{\ast }$ values in both $x$–$z$ and $y$–$z$ planes at $Re=1200$. In contrast to the first three modes, the sphere showed significant motion in the streamwise and the lateral directions as well. As can be seen, the regularized trajectory observed at $U^{\ast }=13.9$ (mode II) became irregular as the sphere transitioned to mode IV. Moreover, the sphere followed a random trajectory with a large amplitude in the transverse ($z$) direction. However, the dominant sphere oscillation frequency was close to the system’s natural frequency. In this mode, no increment was found in the time-mean drag coefficient (figure 8) and the fluctuation force components were small in magnitude. There was no correlation between forces and the sphere vibration. In addition, the wake was irregular in strength and frequency, with several vortex loops formed per oscillation cycle (see figure 25).

Figure 25.

Wake structure observed in mode IV ($Re=1200$, $m^{\ast }=0.8$, $l^{\ast }=10$ and $U^{\ast }=30$) visualized with isosurfaces at $Q=0.001$. Flow from left to right.

To investigate the nature of the mode IV response, the phase plots and Poincaré maps were used. Figure 26 shows the variation of phase plots of transverse velocity versus transverse displacement of the sphere, as the reduced velocity increased, leading to a transition from mode II to mode IV. At $U^{\ast }=10$ (mode II), the phase plot was a circular orbit representing a highly periodic sphere response. As $U^{\ast }$ increased, the phase plot gradually became more disordered. This indicates a gradual transition from mode II to mode IV. Figure 27 shows the Poincaré maps generated by each upward zero crossing of the normalized sphere displacement for the same $U^{\ast }$ values as in figure 26. The Poincaré maps are concentrated at the origin for $U^{\ast }=10$ and $12$ (mode II). Poincaré maps start to spread linearly as $U^{\ast }$ increased up to $18$, and beyond that they spread radially (mode IV). When the points in Poincaré maps are spread widely, it indicates that the dynamical system is exhibiting a chaotic behaviour. Thus, the mode IV response observed at higher reduced velocities can be classified as a chaotic response.

Figure 26.

Phase plots of transverse displacement, $Z$, for various reduced velocities: $U^{\ast }=10$, $12$, $14$, $16$, $18$, $20$, $27$ and $32$, at $Re=1200$ and $m^{\ast }=0.8$.

Figure 27.

Poincaré maps for $Z/D$, taken from each upward zero crossing of the sphere transverse displacement for various reduced velocities: $U^{\ast }=10$, $12$, $14$, $16$, $18$, $20$, $27$ and $32$, at $Re=1200$ and $m^{\ast }=0.8$.

For a static sphere, Brücker (Reference Brücker2001) measured broad low frequencies for the streamwise vortex formation. Therefore, the motion of the sphere seems to originate from the wake pattern of the sphere even in the mode IV regime. The sphere is likely to exhibit a chaotic motion rather than a periodic motion at these higher reduced velocities since the flow speed is comparatively higher. As discussed in the previous subsection, when the mass ratio increased from $0.8$ to $80$, mode III appeared before mode IV. Here, we can see that the inertia of the sphere plays a major role in this low-frequency regime. When the inertia is high, then it tends to show mode III characteristics before mode IV, but when it is low, it shows mode IV behaviour.

For light spheres, modes III or IV were not observed experimentally due to experimental limitations (Jauvtis et al. Reference Jauvtis, Govardhan and Williamson2001). The maximum $U^{\ast }$ considered in those experiments was $U^{\ast }\approx 20$. Jauvtis et al. observed mode II with a constant amplitude up to the highest $U^{\ast }$ value they considered. Mode II might be continued even for larger $U^{\ast }$ values. Since mode IV is a chaotic motion, it is likely to appear for light spheres in the turbulent regime.