3.1. The overall behaviour of the hydrofoil

Figure 2 shows how the dominant frequency of the hydrofoil's oscillations as well as its amplitudes and mean values of displacements in the crossflow (CF) and inline (IL) directions change with increasing reduced velocity. The frequency ratio and the reduced velocity are defined as $f^*=f/f_n$ and $U^*=U/f_n c$, respectively, where $U$ is the incoming flow velocity and $f_n$ is the hydrofoil's natural frequency in air. We use the natural frequency in air in these definitions because the free decay response of the hydrofoil in water was overdamped. Two lines are superimposed on the data corresponding to the expected frequency of vortex shedding from the upstream cylinder, $f_s$, and half of that, $f_s/2$, assuming a constant Strouhal number of $St=0.2$. Four distinct regions of behaviour are identified and are depicted in figure 2 separated by vertical dotted lines. Sample snapshots of flow visualization are also presented in figure 2 to give an overall view of the flow around the hydrofoil. We discuss these visualizations with details later in this section.

Figure 2.

Summary of the hydrofoil's response in the wake of a fixed cylinder. (a) Peak frequencies of the hydrofoil's oscillations in the CF and IL directions versus reduced velocity. Dashed lines indicate the shedding frequency and half shedding frequency calculated using $St=0.2$. Frequencies are normalized by the natural frequency in air, $f^*=f/f_n$. (b) Normalized amplitude of oscillations, $A/c$, in the CF and IL directions, together with (c) the normalized mean values of the CF and IL displacements, $\bar {A}/c$. The displacements are measured at a point located at 0.25$c$ from the leading edge of the hydrofoil. Sample snapshots of flow visualizations are shown for cases in (d) region I and (e) region III.

For $U^*<1.4$ (region I) the oscillation frequencies in both the IL and CF directions closely follow the shedding frequency, implying a synchronization between the IL and CF oscillation frequencies and the shedding frequency. For $1.4< U^*<2.1$ (region II), the oscillation frequencies deviate from the Strouhal line and eventually reach the half-Strouhal line at the end of this region. This region can be considered a transition region, where the IL frequency moves directly to the half-Strouhal frequency, while the CF frequency initially deviates from the Strouhal frequency slightly, and then moves quickly to the half-Strouhal frequency towards the end of this region. In regions I and II, the amplitudes of oscillations in both the CF and IL directions initially increase for increasing $U^*$, then decrease and again increase, reaching the maximum amplitudes in each direction at the end of region II. At the end of the transition region, the oscillation frequencies in the CF and IL directions become equal and follow the half-Strouhal line for a reduced velocity range of $2.1< U^*<2.7$ (region III). In this region, both the CF and IL amplitudes stay more or less constant. Finally, for $U^*>2.7$, the frequencies approach zero (region IV), and small-amplitude oscillations are observed as the hydrofoil is moved to the side. The mean values of the CF and IL oscillations stay close to 0 and 4.5$D$, respectively, for all reduced velocities for which oscillations are observed (figure 2c). At the highest reduced velocity, the mean values increase in both directions. Visual inspection of the wake based on bubble visualizations indicates that the hydrofoil has moved to a side at this reduced velocity and does not interact with the vortices that are shed in the wake of the cylinder.

3.2. Sample time histories

Sample time histories of these four regions together with their corresponding fast Fourier transform (FFT) plots are shown in figure 3. As was observed in figure 2(a), a transition from a 1:1 ratio between the CF and IL oscillation frequencies in region I to another 1:1 ratio in region III is clear in the FFT plots of figure 3. The 1:1 ratio in region I is synchronized at the Strouhal frequency. Then in region II, the frequency contents have contributions at the Strouhal frequency and frequencies lower than the Strouhal frequency, and in region III the main contribution is switched to the half-Strouhal frequency, with small but present contributions from the frequencies at the Strouhal frequency mainly in the CF direction. In region IV, the oscillations are clearly at a very low frequency compared with the previous cases.

Figure 3.

Sample cases of the hydrofoil's response in the wake of a fixed cylinder. Time histories and FFT plots of the IL, $x^*=x/c$, CF, $y^*=y/c$, and torsional, $\theta$, directions for (a) $U^*=1.2$, (b) $U^*=1.7$, (c) $U^*=2.4$ and (d) $U^*=3.1$. The time histories are plotted against reduced time $t_r = t U / c$.

In region I, the amplitudes of oscillations in both directions fluctuate synchronously. In region II, the IL amplitude is considerably smaller than the CF amplitude, and clear signs of a frequency lower than the shedding frequency are observed in the CF displacement time history. In the sample time histories of region III, oscillations with half of the shedding frequency are the main observed oscillations, with frequencies equal to the shedding frequency also observed in the CF direction. In region IV, oscillations of very low frequency are observed in both directions. Within this region, the hydrofoil moves to the side and remains mainly static.

3.3. The observed trajectories

By visually inspecting the hydrofoil's trajectory in each cycle of its oscillations, we have identified seven distinct types of trajectories that the hydrofoil follows at different reduced velocities. In some cases the hydrofoil follows the same trajectory over several cycles of oscillations, and in some cases the trajectories are changed after only a few cycles. Figure 4 shows the histograms of occurrence of these trajectories, together with samples of the trajectories and the frequency contents observed for each type of trajectory.

Figure 4.

Different trajectories that are observed in the response of the hydrofoil in the wake of a fixed cylinder. (a) The rate of their occurrence, (b) the trajectory's shapes and (c) FFTs of their displacements in the IL, $x$, and CF, $y$, directions.

From these plots, it becomes clear that in region I, where the frequencies of the IL and CF displacements are equal and follow the Strouhal frequency, teardrop, linear and oval trajectories are the main trajectories that are observed. In the transition region (region II), the three-point and figure-eight trajectories appear in the time histories. The figure-eight trajectories become the dominant ones towards the end of the transition region, which corresponds to the range where a 2:1 ratio between the frequencies in the CF and IL oscillations is observed in the frequency plots of figure 2. The figure-eight trajectories continue to occur in region III, due to the small contribution of the frequency at the Strouhal value in the CF displacements, and together with the chevron trajectories are the most commonly occurring trajectories in this region.

A map of the observed trajectories in each cycle of oscillations versus the trajectory in the preceding cycle is given in figure 5 for trajectories observed in all reduced velocities tested here. The magnitude shows the likelihood of observing the trajectory that is observed in any cycle in the cycle that follows. A larger value on the $45^{\circ }$ angle line corresponds to a higher likelihood of a repeated trajectory. It is clear from this plot that linear, oval, chevron and figure-eight trajectories occur over longer periods of oscillations, and once the system starts following one of them, it follows the same trajectory over several cycles, while teardrop, three-point and five-point trajectories occur more during transition from one type of trajectory to another one, and do not last long. Clearly, the exact trajectories that the hydrofoil follows can change if the relative size of the hydrofoil and the cylinder changes. In our experiments, the cylinder diameter was slightly smaller than the hydrofoil's chord. A larger cylinder would have resulted in shedding of larger vortices in the wake, which could have influenced the hydrofoil differently.

Figure 5.

A map of the observed trajectory in each cycle versus the trajectory observed in the preceding cycle in the response of the hydrofoil in the wake of a fixed cylinder.

3.4. Flow visualizations

In this section we discuss the details of how the vortices that are shed in the wake of the fixed cylinder interact with the hydrofoil, by focusing on two sample cases from regions I and III of the response. Figure 6 shows snapshots of a case from region I in which the hydrofoil follows an oval trajectory, and figure 7 shows snapshots of a case from region III in which the hydrofoil follows a figure-eight trajectory. In both figures, the interactions between the vortices and the hydrofoil are presented in a series of 12 snapshots of the wake, which correspond to one complete cycle of oscillations. The IL and CF oscillations as well as the trajectories are shown in the figures, with numbers that correspond to the visualization snapshots. In all these snapshots, the two vortices that are shed from the cylinder are clearly visible with the colour red corresponding to the positive (clockwise, CW) vortex and the colour blue corresponding to the negative (counterclockwise, CCW) vortex.

Figure 6.

The interactions of the hydrofoil with the vortices in the wake of the fixed cylinder in a sample case of region I, together with the IL and CF time histories and the hydrofoil's trajectory. See supplementary movie 1 available at https://doi.org/10.1017/jfm.2021.539 for an animation of the bubble image velocimetry (BIV) results.

Figure 7.

The interactions of the hydrofoil with the vortices in the wake of the fixed cylinder in a sample case of region III, together with the IL and CF time histories and the hydrofoil's trajectory. See supplementary movie 2 for an animation of the BIV results.

In snapshot 1 of figure 6, the hydrofoil interacts with the negative (CCW) vortex. The presence of the negative vortex causes the hydrofoil to move in the positive IL direction (the direction of flow) and upward in the CF direction up to the hydrofoil's maximum amplitudes in both directions (snapshots 2 and 3). After rounding the top of the vortex, the hydrofoil moves downward and upstream (snapshot 4), and starts its interaction with the positive (CW) vortex. This vortex then causes further displacement of the hydrofoil in the same directions. In snapshots 5–7, the hydrofoil moves downward in the CF direction and downstream in the IL direction as it is driven by both the CW and the CCW vortices. In snapshots 8–10, the CW vortex is deflected away from the hydrofoil, and the hydrofoil rotates in the CW direction and aligns itself with the direction of the incoming flow momentarily, before it returns to the same position as snapshot 1 in snapshot 12, and repeats the same cycle.

In the first two snapshots of figure 7, the hydrofoil interacts with the CW vortex, and moves downward in the CF direction and upstream in the IL direction (snapshot 3). Subsequently, the hydrofoil is at its lowest CF point in snapshot 4. The most striking difference in this case compared with the sample case from region I is that the hydrofoil traverses below the CCW vortex (snapshots 5 and 6) before again interacting with the next CW vortex. In interacting with the next CW vortex, snapshots 7 and 8, the hydrofoil is again within the wake of the cylinder, and in snapshots 9–12 the hydrofoil rotates with the CCW vortex and is driven to follow a path between the CW and CCW vortices before returning to the same initial angle and CF position as snapshot 1.

These wake visualizations at regions I and III clarify the reason why the frequency of the hydrofoil's oscillations changes from a 1:1 ratio at the Strouhal frequency to another 1:1 ratio, but at half the Strouhal frequency as the reduced velocity is increased. At lower reduced velocities (region I), the hydrofoil is affected by both vortices that are shed from the two sides of the cylinder in each cycle of its oscillations, resulting in a fluctuating force with the same frequency as the shedding frequency in both the IL and CF directions. In region III, the hydrofoil interacts with only one of the vortices that are shed in each cycle of its oscillations and misses the second vortex, and as a result the external force that acts on the hydrofoil has a frequency equal to half the shedding frequency.

4.1. The hydrofoil's displacement

Figure 9 shows how the mean and fluctuating amplitudes of the hydrofoil's oscillations in the CF and IL directions change versus the rotation rate. It is clear from figure 9(a) that the mean displacements of the hydrofoil in the CF and IL directions increase with $\alpha$ until $\alpha \approx 2.4$. For higher values of $\alpha$, the mean value of the CF displacement stays more or less constant, but the mean value of the IL displacement decreases first and then stays constant for $\alpha >4$. Oscillation amplitudes are relatively large initially for no rotation ($\alpha =0$) (figure 9b). As the rotation is applied, the oscillation amplitudes in both the CF and IL directions decrease. By $\alpha =2.4$, the oscillation amplitudes in both directions are smaller than the mean values of displacements in those respective directions. For higher rotation rates, a slight increase is observed in the CF amplitude oscillations, but no major change is observed in the IL amplitudes, both staying much smaller than the mean values for all rotation rates larger than $\alpha =2.4$.

Figure 9.

The hydrofoil's response in the wake of a cylinder forced to rotate in one direction. (a) Mean and (b) fluctuating values of the CF and IL displacements of the hydrofoil versus the rotation rate, $\alpha$. Vertical dashed lines at $\alpha = 2.4$ and $\alpha = 4$ highlight the three regions of response.

The response of the hydrofoil can be related to the wake that is observed for the cylinder. It has been shown previously (Mittal & Kumar Reference Mittal and Kumar2003; Seyed-Aghazadeh & Modarres-Sadeghi Reference Seyed-Aghazadeh and Modarres-Sadeghi2015) that for a cylinder forced to rotate in one direction, such as the cylinder that we have used in the present work, an increase in the rotation rate results in the suppression of vortex shedding at $\alpha =2.4$. This value corresponds to the critical value for $\alpha$ observed in the amplitude plots of figure 9 here, which marks the end of an increase in the mean value of displacement in both the CF and IL directions and a decrease of the amplitude of oscillations in both directions. The decrease in the amplitude of oscillations is due to the suppression of shedding in the wake of the upstream cylinder. Overall, for $\alpha <2.4$ the hydrofoil's oscillations are due to the shedding of vortices that are observed in the wake of the cylinder. As the rotation rate increases, the amplitudes of these oscillations decrease until $\alpha =2.4$, where the vortex shedding is suppressed and as a result the amplitudes of oscillations decrease significantly. According to Mittal & Kumar (Reference Mittal and Kumar2003), at around $\alpha =4$, a large vortex forms in the wake of the rotating cylinder. This large vortex is formed around 5$D$ downstream the cylinder, which is approximately where the hydrofoil is located in the present experiments. The formation of this large vortex is due to the fact that at rotation rates larger than $\alpha =4$, the shear layers leave the cylinder on one side. We have also observed this change in the location where the shear layers leave the cylinder in our flow visualization, and as a result, a change in the response of the hydrofoil at approximately $\alpha =4$. Our flow visualizations reveal the differences in the wake of the cylinder in the three different regions identified in figure 9, as discussed in the following section.

4.2. Flow visualizations

In figure 10(a), the shedding of vortices in the wake of the cylinder is clear. It is the interaction of these vortices with the hydrofoil that provides the oscillatory motion of the hydrofoil in region I. In region II (figure 10b), the vortices are suppressed, and the flow is deflected upward, causing the hydrofoil to move to one side. In region III, while the shedding of vortices similar to what we observe in region I is not observed, the large rotation rate has caused the boundary layer from the lower side of the cylinder to move all the way to its upper side, resulting in a separation point on the upper side of the cylinder, and a major deviation of the flow direction from the direction of the incoming flow.

Figure 10.

Interactions of the hydrofoil with the wake of the cylinder rotating in one direction shown through the vorticity plots and streamlines in the three regions observed for the response of the hydrofoil: (a) $\alpha =0.9$, region I, (b) $\alpha =2.8$, region II and (c) $\alpha =4.7$, region III.

5.1. Amplitude and frequency of response

Figure 12 shows how the peak oscillation frequencies observed in the CF and IL directions change as the forcing frequency (and as a result the shedding frequency in the wake of the cylinder) is varied. In this plot, the oscillation frequency is normalized by the natural frequency in air, $f^*=f/f_n$. Trajectories of the hydrofoil response are also shown in the figure. At lower cylinder frequencies, a 2:1 ratio between the IL and CF oscillation frequencies is observed. Within this range, the trajectories follow a figure-eight pattern, and the hydrofoil moves upstream when it is at the extremes of its CF displacement (the direction of arrows in the figure). The IL amplitude is initially very large (${\approx }0.9c$) at approximately 50 % of the CF amplitude, resulting in wide figure-eight trajectories. As the forcing frequency is increased, while the 2:1 ratio between the IL and CF oscillation frequencies remains, the phasing between the IL and CF displacements changes such that the figure-eight trajectories resemble a crescent shape. Eventually, the IL amplitude decreases and the figure-eight trajectories become mainly vertical trajectories. This is then followed by a loss of the 2:1 ratio, which is replaced by a 1:1 ratio between the IL and CF displacement frequencies, resulting in trajectories that are mainly in the CF direction, and for higher forcing frequencies, trajectories that no longer follow a repeatable pattern. It is observed that as the forcing frequency is increased, both the CF and IL amplitudes decrease slightly in the range where a 2:1 frequency ratio is observed; however, they both are larger than their respective values in the range where the 2:1 ratio is lost. The CF amplitude decreases to values half of the initial values for higher forcing frequencies, while the IL amplitude goes through a relatively small, but noticeable decrease. The fact that a 2:1 ratio between the IL and CF oscillations results in larger amplitudes of oscillations in both directions has been observed and discussed before in studies of vortex-induced vibration (VIV) of a two-degree-of-freedom (2 DOF) cylinder in flow, a model problem in fluid–structure interactions (Dahl, Hover & Triantafyllou Reference Dahl, Hover and Triantafyllou2006; Dahl et al. Reference Dahl, Hover, Triantafyllou, Dong and Karniadakis2007, Reference Dahl, Hover, Triantafyllou and Oakley2010). Based on these studies, the 2:1 ratio is the desired ratio for a 2 DOF cylinder in flow: the shedding of vortices in the wake exerts forces at two times the shedding frequency in the IL direction and equal to the shedding frequency in the CF direction, and as a result the cylinder tends to follow a 2:1 ratio of oscillations more easily, and thus exhibits larger amplitudes of oscillations. The observation in the present work is very similar: the vortices that interact with the hydrofoil exert forces at two times the shedding frequency in the IL direction and equal to the shedding frequency in the CF direction, and as a result the hydrofoil exhibits a 2:1 frequency ratio between its IL and CF oscillations. Sample snapshots of the wake visualizations in the two regions of response (region I with 2:1 frequency ratio and region II with 1:1 frequency ratio) are given in figure 12(c,d). We will give more details of the mechanism by which these vortices interact with the hydrofoil when we discuss the flow visualization of this case.

Figure 12.

The hydrofoil's response in the wake of a cylinder forced to rotate periodically. (a) Predominant frequency in the CFand IL directions versus the forcing frequency together with samples of trajectories. The two lines in the plot represent a 1:1 ratio between the dimensional output frequency and the dimensional input frequency (the lower line) and a 2:1 ratio (the upper line). (b) The amplitudes of oscillations in the CF and IL directions versusthe forcing frequency. Sample snapshots of flow visualizations for (c) region I and (d) region II.

Figure 13 shows sample time histories of the CF and IL displacements of the hydrofoil for five different forcing frequencies. For the first two cases, where a figure-eight trajectory is observed in figure 12, the 2:1 ratio between the IL and CF oscillation frequencies is clear. In the CF frequency contents, besides the peak frequency that occurs at the forcing frequency, contributions at two times and three times the fundamental frequency are also observed. The contributions of these high-frequency contents are also visible in the amplitude plots of these CF displacements. Higher harmonics have been observed previously in the VIV response of a 2 DOF cylinder (Dahl et al. Reference Dahl, Hover, Triantafyllou and Oakley2010). In the VIV response of a cylinder, the presence of the second harmonic is attributed to the asymmetries in the wake (Du, Jing & Sun Reference Du, Jing and Sun2014) and the presence of the third harmonic is related to the shedding of triple vortices in the wake (Dahl et al. Reference Dahl, Hover, Triantafyllou, Dong and Karniadakis2007), or the angles that the fluctuating flow forces make with the CF and IL directions (Seyed-Aghazadeh et al. Reference Seyed-Aghazadeh, Carlson and Modarres-Sadeghi2017). In figure 13(c), a small second-harmonic frequency is still observed in the IL direction. This small peak disappears for higher forcing frequencies. Figure 13(d) shows a sample of cases where the motion is mainly in the CF direction, with a very small amplitude in the IL direction. At higher forcing frequencies (figure 13e), oscillations in both the CF and IL directions become of much lower frequency than before and occur at half the forcing frequency. This example corresponds to a case where the trajectory does not follow any repeatable pattern as shown in figure 12.

Figure 13.

Time histories and their corresponding FFTs of the hydrofoil's response for various forcing frequencies of the upstream cylinder: (a) $f^*_{cyl}=0.06$, (b) $f^*_{cyl}=0.08$, (c) $f^*_{cyl}=0.10$, (d) $f^*_{cyl}=0.125$ and (e) $f^*_{cyl}=0.17$.

The third harmonic that is observed in the CF oscillations is even more significant in the torsional motion of the hydrofoil. Figure 14 shows the time histories for a sample case where a figure-eight motion has been observed. Clearly, the third-harmonic peak in the torsional direction is almost as large as the first-harmonic peak. This large contribution of the third harmonic is also clear in the time history of the torsional motion. It is worth mentioning that in previous studies of 2DOF VIV of a cylinder where the third harmonics had been observed, they had been observed mainly in the flow forces, and their contributions in the displacement frequencies were very small. Here, however, we observe very large third-harmonic contributions in the displacement time histories. The large third-harmonic component in the torsional oscillations of the hydrofoil explains the source of this third harmonic. When the hydrofoil goes across the wake, its angle of attack is at its maximum, resulting in the hydrofoil moving almost perpendicular to the direction of the incoming flow. Then as the hydrofoil reaches its maximum CF displacement, it rotates and its angle of attack decreases, enabling the hydrofoil to travel upstream in a direction almost parallel to the direction of the incoming flow. As the hydrofoil levels out after traversing the wake, it undergoes one cycle of small-amplitude oscillations about the zero angle of attack, as shown in figure 14(c). After this small-amplitude oscillation, the hydrofoil continues its motion in the CF direction towards the other extreme of the CF amplitude, therefore creating a third-harmonic component in the frequency of its oscillations.

Figure 14.

The (a) IL, (b) CF and (c) torsional displacements together with their corresponding FFT plots for the hydrofoil's response in the wake of a cylinder that is forced to rotate at a frequency of $f^*_{cyl}=0.06$.

As the forcing frequency is increased, there is not enough dwell time between the shed vortices, and as a result the hydrofoil cannot undergo the small-amplitude oscillation about its torsional degree of freedom when it reaches the extremes of its CF oscillations, and the third harmonic in the CF direction disappears. This occurs around $f^*_{cyl}=0.09$, where the loops of the figure-eight trajectory collapse and the trajectory becomes C-shaped. Increasing the forcing frequency further reduces the dwell time between shedding of vortices even more, such that the hydrofoil cannot recover its upstream motion at the extremes of its CF oscillations, and therefore the C-shaped trajectory is flattened and by $f^*_{cyl}=0.11$ the trajectory is mainly in the CF direction only. A sample of a case where the CF, IL and torsional oscillations all exhibit the same peak frequency is shown in figure 15 for $f^*_{cyl}=0.125$. This trend holds up to an external frequency of $f^*_{cyl}=0.14$, where the trajectory loses its coherence.

Figure 15.

The (a) IL, (b) CF and (c) torsional displacements together with their corresponding FFT plots for the hydrofoil's response in the wake of a cylinder that is forced to rotate at a frequency of $f^*_{cyl}=0.125$.

5.2. Flow visualizations

A series of snapshots of the cylinder's wake when a figure-eight trajectory is observed are shown in figure 16. These snapshots are for the case of $f^*_{cyl}=0.06$, and they cover one complete cycle of oscillations. Snapshot 1 in the figure corresponds to the time in the cycle when the hydrofoil crosses the zero CF amplitude as it moves upward. At this instant, a large CW vortex is observed downstream of the hydrofoil, and the hydrofoil is attracted to this vortex and moves towards it, and as a result it moves both upward and to the right. As it reaches its maximum CF displacement, the hydrofoil rotates and starts moving downward (snapshots 2 and 3). At the same time, a CCW vortex is shed from the lower side of the cylinder and moves towards the hydrofoil (snapshot 4). The hydrofoil rides on top of this vortex on its way down (snapshot 5), and the vortex passes on the lower side of the hydrofoil and travels to its wake (snapshot 6). Then the CCW vortex is on the lower right side of the hydrofoil (snapshot 7) and the hydrofoil is attracted to it and as a result moves to the right again (snapshots 8 and 9). When the CCW vortex is far from the hydrofoil, the hydrofoil stops its motion in the positive IL direction, rotates and moves upward (snapshots 10 and 11). At the same time, a CW vortex is shed from the upper side of the cylinder (snapshot 12). This CW vortex moves towards the hydrofoil. The hydrofoil moves around it (snapshot 13), and the CW vortex goes to the top right of the hydrofoil (snapshot 14), and the hydrofoil is attracted to it, and this brings the hydrofoil to the beginning of its cycle (snapshot 1). These visualizations explain the nature of the 2:1 frequency ratio that has been observed in the response of the system. The two vortices that are shed from the cylinder force the hydrofoil to go to the extremes of its IL oscillations twice over one cycle of its oscillations in the CF direction. In essence, this is very similar to the reason a 2:1 ratio is observed in the frequency response of a 2 DOF cylinder undergoing VIV: the shedding frequency in the IL direction is twice that in the CF direction, and as a result a 2:1 ratio is observed. Here, the two vortices that are shed from the two sides of the cylinder have symmetric influences on the hydrofoil in the CF direction, but identical influences in the IL direction.

Figure 16.

A series of BIV wake visualization snapshots for a single period of the hydrofoil's response in the wake of a cylinder that is forced to rotate at $f^*_{cyl}=0.06$, when a figure-eight response is observed. See supplementary movie 3 for an animation of the BIV results.

Figure 17 shows a series of snapshots of the wake visualization for a case where a 1:1 frequency ratio between the IL and CF oscillations is observed. These snapshots correspond to a case with $f^*_{cyl}=0.125$. In snapshot 1, the hydrofoil is located at its zero CF amplitude location and moves upward. At this instant, a CW vortex is observed in the wake of the hydrofoil and a CCW vortex is approaching the hydrofoil, resulting in a hydrofoil that is encompassed by two counter-rotating vortices located at approximately the same vertical locations. As a result, the overall force that acts on the hydrofoil from these two vortices in the IL direction is negligible, and the main flow force is in the CF direction. The hydrofoil then travels all the way to the upper extreme of its CF amplitude (snapshots 2 and 3), then it rotates (snapshot 4) and rides on the CCW vortex that was shed from the lower side of the cylinder (snapshots 5 and 6). As the hydrofoil finds its way around the CCW vortex, another CW vortex is already shed from the upper side of the cylinder and interacts with the hydrofoil (snapshot 7) at approximately the same vertical location as the previous two vortices (i.e. at a location of $y \approx 0$). Then the hydrofoil finds itself between two counter-rotating vortices again (snapshot 8) and travels downward to the lower end of its CF oscillations (snapshot 9), where it rotates and starts its movement upward (snapshots 10 and 11). At the same time, a second CCW vortex is shed from the lower side of the cylinder, and the hydrofoil finds itself between two counter-rotating vortices, rotating in the same directions as the vortices observed in snapshot 1, and the loop continues.

Figure 17.

A series of BIV wake visualization snapshots for a single period of the hydrofoil's response in the wake of a cylinder that is forced to rotate at $f^*_{cyl}=0.125$, when a linear response is observed. See supplementary movie 4 for an animation of the BIV results.

The differences in the wake of the cylinder in the two cases are clear in figures 16 and 17. As the forcing frequency is increased, the frequency of shedding of vortices in the wake increases. Besides the increase in the frequency, the width of the wake decreases and the vortices are shed closer to the centre of the wake at higher forcing frequencies. This decrease in the width of the wake with increasing forcing frequency has also been observed by Tokumaru & Dimotakis (Reference Tokumaru and Dimotakis1991). The combination of these two effects results in the change in the type of trajectories that we have observed. For lower frequencies, where the vortices are shed at lower frequencies and at the upper right and the lower right of the cylinder, the snapshots of figure 16 show how the hydrofoil interacts mainly with one vortex at any given time, and is influenced by that vortex. For higher forcing frequencies, however, the vortices are shed at higher frequencies and at the centre of the wake, causing the hydrofoil to interact with two vortices simultaneously at any given time as is clearly observed in snapshots of visualizations in figure 17. This interaction with two vortices at the same time then results in the same frequency of oscillations in both the CF and IL directions in the response of the hydrofoil.