3.3.1. Cylinder oscillation patterns

Time traces of the cylinder position are shown in figure 4 with the spanwise surface wire in question at different angular locations. In all plots, the horizontal axis depicts a total of 30 s in non-dimensional form ${t^\ast } = tU/D$ for visual clarity of the oscillation patterns, although the acquired displacement signal of the cylinder was longer (180 s). The vertical axis in the plots shows the position of the cylinder in non-dimensional form ${y^\ast } = y/D$. Here, ${y^\ast } = 0$ in the vertical axis corresponds to the equilibrium position of the model in still water, and the positive ${y^\ast }$ direction shows cylinder displacements in the direction of the wire-fitted side of the cylinder. Based on the characteristics of the oscillation motion, the wire angular position is divided into eight angular ranges in figure 4.

Figure 4. Time traces of the cylinder position for the application of the wire at different angular locations on the cylinder surface. The axis of angular values, where the range divisions are shown, is not to scale.

For the wire angular positions that fall in range I ($0^\circ \le \theta < 47^\circ $), the position around which the cylinder oscillates shifts toward the wire-fitted side of the cylinder while the cylinder displacements follow a regular sinusoidal pattern. In figure 4, $\theta = 35^\circ $ is provided as a representative case from this first range. In range II ($47^\circ \le \theta < 51^\circ $), the regular sinusoidal response of the cylinder motion is disrupted intermittently by large-amplitude displacements. In figure 4, the oscillation pattern for $\theta = 48^\circ $ is given as a sample case from this range. Because the cylinder response in this second range of wire positions is a combination of low-amplitude sinusoidal and high-amplitude disturbed motions that appear randomly, this angular range II can be considered a critical range for wire placement. As $\theta $ increases within this range starting from the wire position of $\theta = 47^\circ $, the disturbed motion appears more and more frequently, and eventually gives way to another pattern starting at $\theta = 51^\circ $. For range III ($51^\circ \le \theta < 60^\circ $), a non-sinusoidal but distinct oscillation pattern, repeating at every period, is observed, as depicted for the sample case of $\theta = 53^\circ $ in figure 4. Starting from $\theta = 60^\circ $, the cylinder with the wire in the angular range IV ($60^\circ \le \theta < 70^\circ $) depicts a motion consisting of quasi-sinusoidal displacements along with the sporadic appearance of irregular oscillation patterns. When the surface wire is placed within range V ($70^\circ \le \theta < 75^\circ $), the cylinder shows a response that is nearly sinusoidal. This pattern is not symmetric around the equilibrium position such that the oscillation pattern has a wider peak for movements on the wire side (for positive values of ${y^\ast }$). In range VI ($75^\circ \le \theta < 105^\circ $), the oscillation pattern has a quasi-uniform character, with the appearance of small-amplitude modulations between cycles. When the wire is placed at any of the angular positions in range VII ($105^\circ \le \theta < 109^\circ $), the amplitude of the cylinder oscillations diminishes substantially. In this seventh range, the largest attenuation in the amplitude of the cylinder motion occurs when the wire is at around $\theta = 107^\circ $, where, as shown in figure 4, the cylinder stays at its equilibrium state with almost no oscillatory motion. For wire positions of range VIII ($109^\circ \le \theta \le 180^\circ $), the oscillatory motion for the cylinder restarts with a nearly sinusoidal pattern, as seen in figure 4 for the representative scenario of $\theta \; = 110^\circ $ from this range, and as the wire approaches $\theta = 180^\circ$, the oscillation pattern resembles more and more the regular sinusoidal pattern of the clean cylinder (with no wire).

3.3.2. Cylinder oscillation frequency

The dominant frequency of cylinder oscillations ${f_d}$ is observed to change significantly depending on the wire angular position $\theta $. This variation is depicted in figure 5, where the dominant frequency of the cylinder motion ${f_d}$ is given in non-dimensional form by normalizing it with the natural frequency of the oscillating system ${f_n}$ in still water (i.e. $f_d^\ast = {f_d}/{f_n}$). The non-dimensional dominant frequency of oscillations for the clean (untripped) cylinder, which corresponds to a value of $f_d^\ast = 1.29$, is also marked in figure 5 with a horizontal dashed line as the baseline to compare with. As discussed earlier, the cylinder depicts a regular sinusoidal motion when the wire is placed in the angular range I $(0^\circ \le \theta < 47^\circ )$. In this range, as the angular position of the wire is increased starting from $\theta \; = 0^\circ $ until approximately $\theta = 15^\circ $, the frequency of the cylinder oscillations remains constant at a value of $f_d^\ast = 1.28$, which is only slightly below $f_d^\ast $ of the clean cylinder. After $\theta = 15^\circ $, the frequency of oscillations in range I gradually increases as the wire angular position increases. When the wire is placed in the angular range II $(47^\circ \le \theta < 51^\circ )$, a single dominant frequency is not discernible for the cylinder oscillations because, as seen in figure 4, the sinusoidal cylinder motion is disrupted by the random appearance of irregular oscillations spanning a wide range of amplitudes and frequencies. For this reason, the second $\; \theta $ range in the $f_d^\ast{-} \theta $ plot, given in figure 5, is filled with a hatch pattern. For angular locations of range III $(51^\circ \le \theta < 60^\circ )$, the displacement motion has two dominant frequencies, as marked in figure 5. Both dominant frequencies in range III increase with an increase in the angular location of the wire placement. The displacement response has one dominant frequency at angular locations in the ranges IV, V and VI, where the frequency of oscillations increases with the wire angle. Here, $f_d^\ast $ shows a sudden increase at $\theta = 75^\circ $, which is the start of range VI $(75^\circ \le \theta < 105^\circ )$. In the angular range VII $(105^\circ \le \theta < 109^\circ )$, there is no dominant frequency because the cylinder does not undergo any discernible oscillation motion (as seen in figure 4). Thus, this range is also covered with a hatch pattern in figure 5. It should be noted here that the hatched areas for the angular ranges II and VII in figure 5 have different interpretations. In the angular range II $(47^\circ \le \theta < 51^\circ )$, the cylinder motion consists of a combination of sinusoidal and irregular motions (as seen in figure 4) encompassing a wide range of amplitudes and frequencies, so a specific frequency cannot be selected as the prevailing one. However, in range VII $(105^\circ \le \theta < 109^\circ )$, the cylinder does not have a discernible oscillation motion, so that a dominant frequency is not detectable. For the angular positions of the wire in range VIII $(109^\circ \le \theta \le 180^\circ )$, a dominant frequency for cylinder oscillations is perceptible again, and figure 5 shows that this frequency is close to the oscillation frequency of the clean cylinder, particularly for $\theta \ge 135^\circ $.

Figure 5. Variation of the frequency of cylinder oscillations in non-dimensional form ($f_d^\ast = {f_d}/{f_n}$) with the wire angular position. The horizontal dashed line indicates the non-dimensional frequency of the oscillation motion for the clean (untripped) cylinder.

To explore the possible temporal variations in the dominant frequency of cylinder oscillations when the surface wire is applied, the short-time Fourier transform (STFT) has been applied to the cylinder displacement data for different wire placement positions. Figure 6 shows the time–frequency spectrogram of the cylinder oscillations, determined via STFT, for the same wire angular locations provided earlier in figure 4. Each angle in figure 6 is representative of the situation seen within one of the eight angular ranges. In these spectrogram plots, the horizontal axis shows the non-dimensional time ${t^\ast } = tU/D$, and the vertical axis shows the non-dimensional displacement frequency $f_d^\ast $ of the wire-fitted cylinder. The colour-coded contour levels indicate the magnitude of the autospectral density at a given frequency and time (i.e. ${S_{{y^\ast }}}({t^\ast },f_d^\ast )$). The colour maps for the amplitude of autospectral density in the spectrogram plots of figure 6 are provided in logarithmic scale to demonstrate the differences in spectral amplitudes for different $\theta $ ranges. Therein, the red regions mark the highest spectral amplitudes, while the blue regions mark the lowest amplitudes. In order to depict a holistic picture of the frequency variation with time, the frequency spectrogram plots presented in figure 6 are shown over a longer period compared with the displacement time traces given earlier in figure 4.

Figure 6. Time–frequency spectrogram of the cylinder oscillations for having the wire at different angular locations on the cylinder surface. The axis of angular values, where the range divisions are shown, is not to scale.

In range I $(0^\circ \le \theta < 47^\circ )$, there is only one dominant value for the frequency $f_d^\ast $, and this frequency is unchanged with time, as apparent from the time–frequency spectrogram of the representative case $(\theta = 35^\circ )$ given in figure 6. For range I, the presence of one constant frequency for the largest spectral magnitudes over the entire time domain is consistent with the regular oscillation motion seen in figure 4. For range II $(47^\circ \le \theta < 51^\circ )$, as seen for the representative case of $\theta = 48^\circ $ in figure 6, the dominant frequency of the cylinder oscillations scatters over a wide range depending on the time. This time-varying nature of the frequency $f_d^\ast $ is consistent with the intermittent switching of the oscillation mode observed in figure 4 for this range. The short periods during which the dominant frequency of the cylinder oscillations $f_d^\ast $ has a fixed value correspond to the times when the cylinder undergoes regular sinusoidal oscillations, and the times when the dominant frequency $f_d^\ast $ is spread over a wide range of values in the spectrogram plot correspond to the times when the cylinder motion becomes irregular, spanning a wide range of amplitudes and frequencies. It should be emphasized here that these characteristics in the time–frequency domain of the sample case $(\theta = 48^\circ )$ are observed for all angular locations within range II $(47^\circ \le \theta < 51^\circ )$.

When the wire is located in range III $(51^\circ \le \theta < 60^\circ )$, as can be observed from the spectrogram plot of the $\theta = 53^\circ $ case in figure 6, two dominant frequencies co-exist at any given time in the displacement signal of the cylinder. This result implies that the cylinder oscillations consist of the superposition of two sinusoidal motions, which is consistent with the displacement pattern presented earlier in figure 4 for this third angular range. For $\theta = 53^\circ $ in figure 6, it should be noted that, in the time–frequency spectrogram of the cylinder oscillations, there exists a few occasions where the dominant frequency $f_d^\ast $ varies over a range of values. During these short time periods, a disturbed motion exists, which occurs randomly within the angular locations of range III.

In range IV $(60^\circ \le \theta < 70^\circ )$, as the spectrogram plot of the $\theta = 60^\circ $ case in figure 6 shows, the predominant frequency of cylinder oscillations shows a slight variation with time within a narrow range of frequencies. However, there is only one dominant frequency at each time step for the displacement motion. These observations comply with the oscillation pattern seen earlier in figure 4 for range IV, which consists of quasi-sinusoidal oscillations with the sporadic appearance of non-sinusoidal patterns. The variation in the dominant oscillation frequency becomes smaller with increasing wire angular location in this range. Consequently, examining the cylinder motion for different angular locations in this range shows that the irregular motion manifests less frequently as the wire angular location increases.

For the next range of angular locations, range V $(70^\circ \le \theta < 75^\circ )$, the time–frequency spectrogram analysis shows continuous contours of very high autospectral magnitudes at one fixed frequency, as seen for $\theta = 72^\circ $ in figure 6. This signifies a regular, coherent oscillation motion for the angular locations of range V. One noticeable feature in the spectrogram plot of $\theta = 72^\circ $ in figure 6 is the appearance of a short-time small increase in the dominant frequency at around ${t^\ast } = 320$, which is accompanied by a temporary drop in the amplitude of the cylinder oscillation. Such frequency variations in range V, however, occur very rarely.

In range VI $(75^\circ \le \theta < 105^\circ )$, the dominant frequency of the cylinder oscillations remains constant, while the spectral magnitude of cylinder displacements at the dominant frequency shows changes with time, as can be seen in the spectrogram plot of $\theta = 86^\circ $ in figure 6. Such fluctuations in the spectral amplitude of the dominant frequency suggest the presence of variations in the coherency of the oscillation motion. The quasi-uniform character of the oscillation pattern with small-amplitude modulations, seen before for range VI in figure 4, complies with this observation.

When the wire is placed at angular locations of range VII $(105^\circ \le \theta < 109^\circ )$, there is no regular oscillation motion, as seen earlier for $\theta = 107^\circ $ in figure 4. As a result, there is no dominant frequency, and the spectral magnitude of the displacements is at the noise level for this range, as observed in the spectrogram plot of $\theta = 107^\circ $ in figure 6.

The time–frequency spectrogram of cylinder oscillations at $\theta = 110^\circ $ in figure 6 reveals the existence of a dominant frequency since the oscillation motion resumes in the angles of range VIII $(109^\circ \le \theta < 180^\circ )$, as depicted in figure 4. Although the spectral magnitude has low values and continuously fluctuates with time at $\theta = 110^\circ $, as the wire is placed at the higher angular locations of range VIII, the spectral magnitude becomes constant with time (not shown in figure 6), and the time traces of the cylinder displacement turn into perfect sinusoids.

To further examine the frequency characteristics of cylinder oscillations, the autospectral density of cylinder displacements ${S_{{y^\ast }}}(\,f_d^\ast )$ was evaluated using the classical FFT analysis. These results are presented in figure 7 for every $1^\circ $ increment of $\theta $ within the ranges I to IV, where the scale of the vertical axis is the same in each plot to ease the comparison of the spectral amplitudes. Figure 7(a) presents the displacement spectra for four angles at the end of range I as a basis for comparison. As mentioned earlier, the cylinder undergoes a uniform sinusoidal oscillation for every wire angular location in range I. It is, therefore, observed that there is only one peak in the spectra of the displacement signal in figure 7(a), and this observation confirms the existence of only one dominant frequency in the first angular range. For all wire angular locations of range II $(47^\circ \le \theta < 51^\circ )$, small spectral amplitudes are distributed over many frequencies, and therefore, no dominant frequency can be selected, as apparent from the spectra provided in figure 7(b). This result complies with the time traces of the cylinder position given in figure 4 and the time–frequency spectrogram of the cylinder oscillations shown in figure 6 for range II, where intermittent switching between seemingly sinusoidal and disturbed cylinder motions was revealed. Although the cylinder fitted with the wire at the angular locations of range II undergoes short periods of somewhat sinusoidal-looking oscillations, a single frequency is still not discernible for this range, as observed in figure 7(b).

Figure 7. Autospectral density of the cylinder displacements, ${S_{{y^\ast }}}(\,f_d^\ast )$, for select locations of the wire in: (a) range I $(0^\circ \le \theta < 47^\circ )$, (b) range II $(47^\circ \le \theta < 51^\circ )$, (c) range III $(51^\circ \le \theta < 60^\circ )$ and (d) range IV $(60^\circ \le \theta < 70^\circ )$.

In the wire angular range III $(51^\circ \le \theta < 60^\circ )$, the co-existence of dual frequencies in the time–frequency spectrogram of the cylinder oscillations, seen in figure 6, agrees with the FFT spectra of the cylinder displacements, given in figure 7(c). For all $\theta $ in figure 7(c), two dominant frequencies are discernible in the spectra of the cylinder displacements. This indicates that the oscillation motion of the cylinder in this $\theta $ range mainly consists of a superposition of two sinusoidal motions with different frequencies, which was also apparent from the time traces of the cylinder position given in figure 4 for the representative case of $\theta = 53^\circ $. This twin-peak phenomenon in cylinder motion derives from the vortex shedding mechanism prevalent in this $\theta $ range, as will be discussed in § 3.4. One interesting observation that comes out of figure 7(c) is that the two dominant frequencies increase with increasing wire angular position. The vertical dashed lines in figure 7(c) mark the minimum and maximum values of these frequencies. Furthermore, as the angular position of the wire increases, the spectra demonstrate a gradual transfer of energy from the larger-frequency motion to the lower-frequency motion, eventually yielding a single-peak spectrum at $\theta = 60^\circ $, which is the start of the fourth range.

In the wire angular range IV $(60^\circ \le \theta < 70^\circ )$, the cylinder displacement data for $\theta = 60^\circ $, given earlier in figure 4, displayed a pattern consisting of both sinusoidal and irregular patterns, which was further supported by the time–frequency spectrogram presented in figure 6 for the same $\theta $. To further elaborate on this interesting observation, the autospectral density of the cylinder displacement is presented in figure 7(d) for varying $\theta $ in range IV. In this range, a slight gradual increase in the dominant frequency of cylinder oscillations is discernible with an increase in the angular position of the surface wire from $\theta = 60^\circ $ to $\theta = 65^\circ $, after which the dominant frequency remains constant. This aspect was also visible in figure 5 for range IV. Furthermore, as the wire angle increases, the small, non-dominant spectral amplitudes gradually decrease while the spectral amplitude of the dominant frequency increases, suggesting lessening in the irregular motion as the wire angular location increases.

3.3.3. The amplitude and mid-position of the cylinder oscillations

The variations of the amplitude and the mid-position of the cylinder oscillations with the angular position of the wire are shown in figures 8 and 9, respectively. In these figures, a polynomial curve is fitted for each range of wire angular positions to demonstrate the overall change. To fit the data within each range to a polynomial function, a standard least-squares algorithm is used. The order of the polynomial function is selected manually. In figure 8 and throughout the study, the amplitude of the cylinder oscillations is defined as half of the distance between the two overall phase-averaged limiting positions that the cylinder sweeps during its oscillations, as explained in § 2, and this amplitude is given in non-dimensionalized form based on the cylinder diameter D, as denoted by ${A^\ast }$ (see (2.1)). The mid-position of the cylinder oscillation $y_{mid}^\ast $ is calculated by averaging the displacement signal, and it is also normalized using the cylinder diameter $(y_{mid}^\ast = {y_{avg}}/D)$.

Figure 8. Variation of the non-dimensional amplitude of oscillations ${A^\ast }$ with the wire angular position $\theta $. The horizontal dashed line indicates the oscillation amplitude of the clean cylinder, which has the value of ${A^\ast } = 0.44$. The uncertainty is the same for all cases and is marked on the plot for some data points as a reference.

Figure 9. Variation of the mid-position $y_{mid}^\ast $ of the oscillating cylinder with the wire angular position. The positive direction of the vertical axis corresponds to the wire side of the cylinder. $y_{mid}^\ast = 0$ is the equilibrium state in still water for both clean and tripped cylinders. The horizontal dashed line indicates the mid-position of oscillation for the clean cylinder. The uncertainty is the same for all cases and is marked on the plot for some data points as a reference.

As can be seen in figure 8, as the wire angle increases within the angular range I $(0^\circ \le \theta < 47^\circ )$, the oscillation amplitude ${A^\ast }$ decreases, attaining its minimum value at $\theta = 46^\circ $, which is 37 % lower than the amplitude of a clean (untripped) cylinder that undergoes VIV under the same flow conditions. Meanwhile, figure 9 depicts that, in this same range $(0^\circ \le \theta < 47^\circ )$, the mid-position $y_{mid}^\ast $ of oscillations increases significantly with the wire angle, which implies that the wire in this range induces a gradually increasing net time-averaged lift force towards the wire side of the cylinder as $\theta $ increases. Remember that the range of the first critical angle for a stationary cylinder exposed to the same flow conditions was $41.2^\circ \le {\theta _{c1}} < 43^\circ $ (as discussed earlier in § 3.1), which falls within the range I of the cylinder undergoing VIV. As discussed earlier, this critical wire location was attributed to the attenuation of the Kármán instability, in the time-averaged sense, for stationary cylinders. However, the cylinder undergoing VIV motion here shows a regular sinusoidal motion at ${\theta _{c1}}$, similar to the results presented in figures 4–6, while its oscillation amplitude is decreased considerably compared with the clean cylinder undergoing VIV, as apparent in figure 8.

In angular range II $(47^\circ \le \theta < 51^\circ )$, figure 8 shows a gradual increase in the amplitude of cylinder oscillations with increasing wire angle. As seen earlier in figure 4, the seemingly sinusoidal oscillation of the cylinder is intermittently disrupted by an irregular motion that involves a range of amplitudes from low to high. However, the high-amplitude irregular motion dwarfs the low-amplitude irregular motion, and that is the reason for the abrupt increase in the amplitude of the cylinder oscillations at the beginning of range II. Due to the presence of both somewhat sinusoidal and irregular motions in this range, two mid-positions can be defined for the cylinder oscillations, as seen in figure 9. The higher values in the mid-position $y_{mid}^\ast $ plot of range II in figure 9 correspond to the near sinusoidal motion, while the lower data points in the same plot represent the mid-position of the oscillations during irregular motion. As the switch of the oscillations between the regular and irregular motions is intermittent for all $\theta $ in this range, the cylinder oscillates sometimes around the upper and sometimes around the lower mid-position given in figure 9 for range II.

The overall inspection of figure 8 shows that, throughout the angular ranges II, III, IV and V, the oscillation amplitude of the cylinder continually increases with the wire angular position, reaching its highest value at $\theta = 73^\circ $ (in the angular range V), which is 102 % higher than the oscillation amplitude of the clean cylinder. Note that attainment of such a large level of oscillation amplitude at this $\theta $ was observed to persist even for a wide range of ${U^\ast }$ values (related result not shown here for brevity). Moreover, figure 9 shows a gradual decrease of the mid-position of the cylinder oscillations with increasing wire angle in the angular range III, and an increase in ranges IV and V.

As seen earlier in figures 4 and 5, both the pattern and the frequency of oscillations change abruptly at $\theta = 75^\circ $ (which is the start of range VI). Likewise, at this $\theta $, abrupt changes are also visible in both the amplitude and the mid-position of the cylinder oscillations from figures 8 and 9, respectively. In the angular range VI, figure 8 depicts that the oscillation amplitude decreases with increasing wire angular position. Interestingly, the negative value of the cylinder mid-position in range VI (in figure 9) implies that the oscillating cylinder is pushed towards its clean side by the effect of the wire (unlike the other ranges). It should be noted that the range of the second critical angle for the stationary cylinder counterpart $(50^\circ \le {\theta _{c2}} \le 80^\circ )$ is a wide range, which starts from the end of range II and extends to the beginning of range VI of the wire-fitted cylinder undergoing VIV. As discussed earlier, the wire at ${\theta _{c2}}$ enhances the Kármán vortex shedding from the stationary cylinder. However, the employment of the wire in the range of the second critical angle of the stationary cylinder has varying effects on the motion characteristics of the oscillating cylinder, depending on the $\theta $ range defined in figure 4.

In the angular range VII, the oscillation amplitude of the cylinder dropped drastically, with the wire at $\theta = 107^\circ $ yielding the minimum value, which is 98 % lower than the amplitude of the clean cylinder (figure 8). At and around $\theta = 107^\circ $, as seen earlier in figure 4, the cylinder is almost motionless, and its mid-position stays near its equilibrium position, as shown in figure 9. This level of amplitude attenuation was observed at around this $\theta $ even over a wide range of reduced velocities. This observation is extremely significant and suggests that something as simple as a single spanwise wire can be used to suppress the VIV of a cylinder almost totally.

The sinusoidal oscillations for the cylinder resume starting from $\theta = 109^\circ $ (which is the start of the angular range VIII). For the wire angular positions in range VIII, figure 8 shows that the amplitude of oscillations increases with increasing $\theta $, approaching asymptotically a finite value of $\; {A^\ast } = 0.61$. Furthermore, inspection of figure 9 shows that the cylinder oscillations in range VIII occur around its equilibrium position.

The characteristics of the cylinder motion discussed so far are expected to be intimately related to the vortex shedding mechanism. To elucidate this aspect, the subsequent section focuses on the flow field characteristics.

3.4.1. Near-wake structure

In this section, to increase insight into the vortex shedding patterns associated with different wire locations, contour patterns of instantaneous vorticity in the near-wake region will be discussed in relation to the cylinder motion for all wire angular ranges. The results presented here aim to reveal the vortex shedding processes featured by the different wire angular ranges, and during this investigation, no assumption is made regarding the phase difference between the fluid forcing and the cylinder motion. In all figures given here, the flow is from left to right, and the wire (not shown in the figures) is located on the upper side of the image of the cylinder. On the contour plots of the instantaneous vorticity, an arrow shown on the image of the cylinder signifies the direction of the cylinder motion at that instant. No arrow is provided whenever the cylinder comes to an instantaneous stop at the beginning or end of an oscillation stroke. Also, in the figures, the time is represented in non-dimensional form ${t^\ast } = tU/D$ and its origin is selected to be the instant when the cylinder is at its lowest point in the particular cycle that is being discussed.

In figure 10, the contours of instantaneous non-dimensional vorticity $\omega D/U$ are shown for the wire at $\theta = 45^\circ $ at instants corresponding to four different cylinder positions selected from one full oscillation cycle of the cylinder. These four cylinder positions are marked on the time trace of the cylinder position, which is also given in the same figure. The wire position $\theta = 45^\circ \; $belongs to angular range I. Inspection of the contour plots in figure 10 reveals the following vortex shedding process at this$\; \theta $: when the cylinder is at position 1 (marked on the ${y^\ast }$ versus ${t^\ast }$ plot), a vortex indicated as ${V_1}$ is formed near the upper side (wire side) of the cylinder. This vortex grows in size and is shed further downstream when the cylinder is at the mid-position of the cycle (position 2). Then, when the cylinder is at its top-most position in this cycle (position 3), another vortex, named ${V_2}$, is formed. ${V_2}$ is significantly stretched and shed in the downstream direction while also moving in the transverse direction when the cylinder reaches position 4, which is a position that comes slightly after the mid-position of this oscillation cycle during the downward stroke. The characteristics of ${V_1}$ and ${V_2}$ are different; ${V_1}$, on the wire side, has a nearly circular shape and travels along the streamwise direction while ${V_2}$ stretches and moves in the transverse direction (in the positive ${y^\ast }$ direction) as it sheds downstream. The observed process of vortex shedding for $\theta = 45^\circ $ may be considered, at first sight, to be similar to the 2S vortex shedding mode, which was revealed by Williamson & Roshko (Reference Williamson and Roshko1988) for clean cylinders undergoing VIV at low reduced velocities. As discussed in their study, in the 2S vortex shedding mode, one vortex is shed during the acceleration of the cylinder at each half of the oscillation cycle, and vortices travel in the streamwise direction in a row. However, in the present case, although there is one single vortex shed per half-cycle, the character of the second vortex is very different from the 2S mode. The second vortex shedding from the smooth side undergoes significant stretching and lateral movement while convecting downstream, which is presumably why the cylinder has been pushed toward the wire side so significantly, as seen in figure 9.

Figure 10. Contours of instantaneous normalized vorticity $(\omega D/U)$ for the wire angular position $\theta = 45^\circ $ (from range I) along with the time traces of the cylinder displacement. The minimum absolute value and the incremental value of contour levels are: ${[|\omega |D/U]_{min}} = 9$ and $\Delta (\omega D/U) = 1.15$.

When the wire is placed at the angular positions of range II, starting from $\theta = 47^\circ $, the sinusoidal motion of the cylinder is intermittently disrupted, as demonstrated earlier in figure 4 for a representative $\theta $ from this range. Figure 11 shows the normalized vorticity contours at four instants selected from different parts of the cylinder's motion for $\theta = 48^\circ $ as an example case from range II. Points 1 and 2, marked on the ${y^\ast }$ versus ${t^\ast }$ plot, are instants from a cycle during the cylinder's sinusoidal motion. Point 1 shows the instant when the cylinder is moving away from its lowest point. This point comes slightly after the start of this upward move. Point 2 is ${\rm \pi} $ away from point 1 in the cycle and corresponds to an instant when the cylinder is moving away from its top-most position. As the cylinder passes from points 1 and 2, vortices ${V_1}$ and ${V_2}$ are formed, respectively, and the vortex shedding process appears to be generally the same as that observed for range I. In this case also ${V_2}$ is stretched and moves in the transverse direction as the cylinder passes the mid-position of the cycle in its downward stroke (which is not shown in figure 11 for brevity). Point 3 is from an instant when the wire interrupts the vortex shedding and introduces irregular cylinder motion. At this point, there are two streams of vorticity coming from each side of the cylinder. Later on, the regular Kármán vortex shedding resumes, as is the case at point 4, creating a new vortex, marked as ${V^{\prime}_1}$, which yields sinusoidal cylinder motion again. From these results, it can be concluded that, in range II, the cylinder switches intermittently between two modes of oscillation (i.e. sinusoidal and irregular motions), which is accompanied by a corresponding switch in the vortex shedding process.

Figure 11. Contours of instantaneous normalized vorticity $(\omega D/U)$ for the wire angular position $\theta = 48^\circ $ (from range II) along with the time traces of the cylinder displacement. The minimum absolute value and the incremental value of contour levels are: ${[|\omega |D/U]_{min}} = 9$ and $\Delta (\omega D/U) = 1.15$.

The switch in the vortex shedding mode of the oscillating wire-fitted cylinder for the angular range II $(47^\circ \le \theta < 51^\circ )$ is remarkably similar to the switch between the irregular shedding and the regular shedding modes first observed by Aydin et al. (Reference Aydin, Joshi and Ekmekci2014) for a stationary cylinder when the wire is at a first critical angle. Recall from § 3.1 that the range of the first critical angle for the stationary cylinder was found to be $41.2 \le {\theta _{c1}} < 43^\circ $ for the wire size and Reynolds number considered in the present study. In the referenced work of Aydin et al. (Reference Aydin, Joshi and Ekmekci2014), it was shown that for the stationary cylinder, the wire at ${\theta _{c1}}$ causes the formation of Kármán vortices to stop for the majority of the time, while the regular vortex shedding resumes only intermittently for short periods of time. For the oscillating cylinder in the present work, the wire at angular range II induces irregularities in the cylinder motion at a wide range of frequencies, as shown in figure 7, and the regular vortex shedding occurs intermittently, as seen in figure 11. To further investigate the similarity in vortex shedding interruption between the stationary and oscillating wire-fitted cylinders, the streamwise velocity signal u was extracted from the PIV data at point $(1.35D,0.56D)$ in the near wake and the corresponding time–frequency spectrogram was obtained from the STFT analysis. The contour plots in figure 12 show these spectrograms for the stationary cylinder fitted with the wire at the first critical angle ${\theta _{c1}} = 42^\circ $ and the oscillating cylinder fitted with the wire at $\theta = 48^\circ $. These plots depict the variation of the normalized frequency of the streamwise velocity $(\,f_u^\ast = {f_u}/{f_n})$ with the non-dimensional time $({t^\ast } = tU/D)$. At times when there is a regular vortex shedding, a specific dominant frequency for flow fluctuations is visible from the high-amplitude contour levels in figure 12. From these spectrograms, it is clear that the vortex shedding process is sporadic for both the stationary and oscillating wire-fitted cylinders in question. Hence, it can be concluded that, for the oscillating cylinder, the effect of wire on disrupting the vortex shedding is shifted to the angular positions of range II $(47^\circ \le \theta < 51^\circ )$, which are higher than the first critical angles of the stationary cylinder $(41.2^\circ \le {\theta _{c1}} < 43^\circ )$.

Figure 12. Time–frequency spectrogram of the streamwise velocity component for: (a) the stationary cylinder with the wire at the first critical angle $({\theta _{c1}} = 42^\circ )$, and (b) the oscillating cylinder with the wire placed at $\theta = 48^\circ $ (within the angular range II). The magnitude of spectra $|S_u^\ast |$ is normalized based on the maximum value in the entire period, and the incremental value of the contour levels is $\mathrm{\Delta [|}S_u^\ast |] = 0.02$.

In the wire angular range III, it has been seen so far that the cylinder motion is the superposition of two sinusoidal oscillations with different frequencies (as discussed in figures 5–8). This observation can be linked to the vortex shedding process, which is depicted in figure 13 for the representative $\theta = 53^\circ $ case from range III. When the cylinder starts its upward motion in the positive ${y^\ast }$ direction (at position 1), a vortex indicated as ${V_1}$ forms. This vortex sheds when the cylinder reaches the mid-position of the cycle (the corresponding vorticity contour is not shown in the figure). Another vortex, ${V_2}$, forms as the cylinder continues to move upwards toward position 2, and this vortex sheds when the cylinder is at the top position of the oscillation pattern. During the acceleration of the cylinder in the negative ${y^\ast }$ direction, a third vortex, ${V_3}$, forms near the wire side of the cylinder (${V_3}$ formation during downward stroke is not shown in the figure). In this case, the wire has induced an additional vortical structure, and the cylinder gains an additional reciprocal motion. As can be seen from the time trace of the cylinder position given in figure 13, for position 3, the cylinder motion diverges starting near the mid-position of the downward stroke of the first sinusoidal motion and ${V_3}$ has already shed when the cylinder starts to move in this positive y* direction. Next, ${V_4}$ is formed from the clean side of the cylinder as the cylinder moves in the negative ${y^\ast }$ direction. Consequently, four vortices are shed per cycle for wire angular range III. This vortex pattern is different from the 2P vortex shedding mode observed during the VIV of a clean cylinder in the lower synchronization range. In the 2P mode, as Williamson & Roshko (Reference Williamson and Roshko1988) discussed, two vortices are shed per half-cycle, and two vortices of opposite sign pair together while travelling downstream. In the vortex pattern of $\theta = 53^\circ $ (representative of range III) given in figure 13, four vortices are shed in total; however, they do not travel together. It is worth mentioning here that, for higher angles in range II, when the cylinder oscillation pattern switches between being sinusoidal and irregular, the vortex shedding mode just discussed for range III is observed occasionally, and as a result, the vortex shedding mode alternates between three modes: the mode observed for range I, the mode observed for range III and the no regular vortex shedding mode (the corresponding results are not shown here for brevity). This switch in the vortex modes in range II is irregular, resulting in a change in the oscillation pattern as the wire angular location increases toward range III.

Figure 13. Contours of instantaneous normalized vorticity $(\omega D/U)$ for the wire angular position $\theta = 53^\circ $ (from range III) along with the time traces of the cylinder displacement. The minimum absolute value and the incremental value of contour levels are: ${[|\omega |D/U]_{min}} = 9$ and $\Delta (\omega D/U) = 1.15$.

The wire at angular locations of range IV induces quasi-sinusoidal oscillations with the random appearance of irregular motions, as can be seen in the time trace of the displacement signal given in figure 14 for $\theta = 60^\circ $ as a representative case from this range. Figure 14 also shows the vortex shedding process for this $\theta $ over two consecutive oscillation cycles. The first oscillation cycle of the cylinder motion that includes positions 1 to 4 is similar to the oscillation motion observed in range III. Accordingly, the vortex shedding process in the first cycle shown in figure 14 for range IV is also identical to the vortex shedding process of range III. As described earlier for figure 13, four vortices are shed in one cycle, and the last two vortices (marked as ${V_3}$ and ${V_4}$ in the vorticity contour plots of positions 3 and 4 in figure 14) induce an additional reciprocating motion on the oscillating cylinder. However, the same vortex shedding process does not always yield this additional reciprocating motion, as seen in the second cycle, which includes positions 5 to 8 in figure 14. At position 5, the first vortex of the cycle, indicated as ${V^{\prime}_1}$, is formed. This vortex moves downstream as the cylinder reaches the end of that upward stroke. The second vortex ${V^{\prime}_2}$ is formed and starts moving downstream as the cylinder changes its direction of motion right before position 6. At position 6, the cylinder moves slowly in the negative ${y^\ast }$ direction. Then, the third vortex ${V^{\prime}_3}$ is shed at position 7. As the cylinder accelerates in the same direction toward position 8, the fourth vortex ${V^{\prime}_4}$ is forming. As can be seen in figure 14, the vortex shedding processes of the second cycle and the first cycle are similar, while the motion of the oscillating cylinder is different. By comparing positions 3 and 7, it is visible that the two vortices, ${V_3}$ and ${V^{\prime}_3}$, look identical, but the direction of motion for the cylinder is opposite. It should be noted that this observed vortex shedding process is repeated in the rest of the oscillation pattern for $\theta = 60^\circ $ as well as at all other angular locations in range IV.

Figure 14. Contours of instantaneous normalized vorticity $(\omega D/U)$ for the wire angular position $\theta = 60^\circ $ (from range IV) along with the time traces of the cylinder displacement. The minimum absolute value and the incremental value of contour levels are: ${[|\omega |D/U]_{min}} = 9$ and $\Delta (\omega D/U) = 1.15$.

In figure 15, the contours of instantaneous normalized vorticity are shown for $\theta = 74^\circ $ as an example case from the wire angular range V. In this $\theta $ range, the cylinder undergoes high-amplitude, nearly sinusoidal oscillations with one dominant frequency. However, the vortex shedding mechanism in range V is significantly different from the other ranges that exhibit sinusoidal cylinder motion. The vortex shedding process in range V appears to be similar to that in ranges III and IV. As the cylinder moves upward (in the positive ${y^\ast }$ direction), a vortex, marked as ${V_1}$ at position 1 in figure 15, forms from the upper side (which is the wire side) of the cylinder. At this position, the vortex marked as ${V_0}$ is from the previous cycle. As the cylinder accelerates to the mid-position of the cycle, ${V_1}$ stretches and moves laterally in the negative ${y^\ast }$ direction. At position 2, the second vortex ${V_2}$ is formed from the clean side of the cylinder. This vortex is moved to a downstream location at position 3. When the cylinder reaches the top-most position in the cycle, the wire induces the third vortex, ${V_3}$, which is seen at position 3; ${V_3}$ travels downstream as the cylinder moves downward (in the negative ${y^\ast }$ direction), as shown in position 4. Furthermore, the fourth vortex ${V_4}$ is formed while the cylinder continues moving downward, as seen at position 4. ${V_4}$ starts to travel downstream when the cylinder reaches the end of the cycle (not shown in the figure). In figure 15, ${V_4}$ at position 4 is equivalent to V₀ at position 1. Accordingly, for the wire angular range V, four vortices are shed per cycle, similar to the ranges III and IV; however, the cylinder motion is different, such that, in range V, the cylinder's direction of motion does not change when the additional vortex (${V_3}$) is induced, but when ${V_3}$ is formed, the cylinder moves more slowly around the top position compared with a sinusoidal motion. A careful inspection of the cylinder oscillation pattern, given in figure 15, shows that the sinusoidal oscillation pattern is not symmetrical. The peaks are wider than the valleys. Although four vortices are shed per cycle for the wire angular range V, the vortex shedding pattern cannot be confirmed as the 2P vortex shedding mode of a clean cylinder because the pairing of vortices in the downstream region of the wake is not visible from the current field of view. On rare occasions, the vortex shedding process in range V is interrupted (not shown in the figure), which causes a temporary drop in the amplitude of the cylinder oscillation accompanied by a short-time small increase in the oscillation frequency; one such instance was seen in the time–frequency spectrogram of $\theta = 72^\circ $ (from range V) in figure 6.

Figure 15. Contours of instantaneous normalized vorticity $(\omega D/U)$ for the wire angular position $\theta = 74^\circ $ (from range V) along with the time traces of the cylinder displacement. The minimum absolute value and the incremental value of contour levels are: ${[|\omega |D/U]_{min}} = 9$ and $\Delta (\omega D/U) = 1.15$.

The subsequent wire angular range is range VI, for which the cylinder depicts a quasi-sinusoidal motion with a lower amplitude and higher frequency in comparison with range V. Range VI starts at $\theta = 75^\circ $, for which the cylinder oscillation pattern and the vorticity contours are presented in figure 16. Similar to the sinusoidal oscillation pattern of range I, two vortices are shed for the wire angular positions within range VI. The first vortex, ${V_1}$ on the wire side, is forming at position 1 (which corresponds to the phase when the cylinder starts to accelerate upwards). At position 2, ${V_1}$ is seen at a downstream location. Next, the second vortex ${V_2}$ is formed from the clean side of the cylinder, as seen at position 3; ${V_2}$ is shedding downstream when the cylinder passes the mid-position of the cycle at position 4. The characteristics of vortices in range VI are different from the previous ranges, such that the vortices are more stretched and the small-scale structures are more discernible in the vorticity contours.

Figure 16. Contours of instantaneous normalized vorticity $(\omega D/U)$ for the wire angular position $\theta = 75^\circ $ (from range VI) along with the time traces of the cylinder displacement. The minimum absolute value and the incremental value of contour levels are: ${[|\omega D/U]_{min}} = 9$ and $\Delta (\omega D/U) = 1.15$.

When the wire is placed in the θ range VII $(105^\circ \le \theta < 109^\circ )$, figure 4 revealed that the sinusoidal oscillation motion of the cylinder ceases, and figure 8 depicted that the amplitude of the cylinder motion reduces by more than 90 % compared with the clean cylinder case. Contour patterns of instantaneous normalized vorticity $(\omega D/U)$ are shown in figure 17 for the cylinder that is fitted with the wire at θ = 107°, which is the case yielding the largest-amplitude reduction in cylinder motion (to be specific, 98 % reduction compared with the clean cylinder). Note that, for the wire angular range VII, experiments were repeated with an extended field of view in the downstream direction to better investigate the observed vortex shedding modes. Two distinct vortex shedding modes are revealed in this range. Figure 17(a) depicts a snapshot from one of these modes, named Mode I here. In this mode, vortices are shed alternately into the wake. Moreover, in this mode, the vortices do not move away from the cylinder centreline as they convect downstream. The second mode, Mode II, is depicted as a snapshot in figure 17(b). In this mode, two vortices that are symmetric about the cylinder centreline form (marked as ${V^{\prime}_1}$ and ${V^{\prime}_2}$). After circulating in their position for a brief while, they shed downstream in the form of small-scale vortices along the centreline of the cylinder. Long-time PIV records of the near wake (for approximately 54 vortex shedding cycles) show that the vortex shedding pattern switches randomly between these two vortex shedding modes (Mode I and Mode II).

Figure 17. Contours of instantaneous normalized vorticity $(\omega D/U)$ for the wire angular position $\theta = 107^\circ $ (from range VII) with random switching of the vortex shedding mode between (a) Mode I and (b) Mode II. The minimum absolute value and the incremental value of contour levels are: ${[|\omega |D/U]_{min}} = 9$ and $\Delta (\omega D/U) = 1.15$.

To further assess the random switch between the two vortex shedding modes (Mode I and Mode II), STFT of the streamwise velocity signal, u, extracted from one point in the near wake of the clean cylinder undergoing VIV motion is compared with that of a cylinder fitted with the wire at $\theta = 107^\circ $. The STFT of the streamwise velocity signal is calculated at point $(1.62D,0.51D)$ for the clean cylinder and at point $(2.13D, - 0.41D)$ for the cylinder having the wire at $\theta = 107^\circ $ to represent the differences discernibly. Figure 18 shows the variation of the spectral amplitude, which is normalized by its maximum value in the entire time period studied. The clean cylinder oscillates with a constant frequency, and its synchronized vortex shedding frequency has a fixed frequency of $f_u^\ast = 1.31$, as apparent from figure 18(a). For the same flow conditions, when the wire is paced at $\theta = 107^\circ $, the vortex shedding frequency varies, as shown in figure 18(b). The high spectral amplitudes (coloured in red based on the colour bar) correspond to the instants where the alternating shedding of vortices occurs (Mode I), and the lower magnitudes of spectra point out to the periods of interrupted vortex shedding (Mode II). Note that Mode I involves the shedding of vortices at a frequency identical to the vortex shedding frequency of a stationary clean cylinder at $Re = {10^4}$ ($f_u^\ast = 2.14$, or in other words, $St\; = \; 0.21$), while Mode II involves shedding of vortices with no predominant frequency. Although here the vortex shedding modes are depicted only for the $\theta = 107^\circ$ case, these modes were equally observed for all $\theta $ in range VII $(105^\circ \le \theta < 109^\circ )$. It can be concluded that, in range VII, the wire achieves attenuation in the VIV response of the cylinder by altering the shedding mode of the vortices.

Figure 18. The short-time Fourier transform (STFT) of the streamwise velocity for: (a) the clean cylinder and (b) the cylinder fitted with the wire at $\theta = 107^\circ $. The magnitude of the velocity spectra is normalized based on the maximum value in the entire period. The normalized frequency resolution is 0.02 and the temporal resolution is 0.21 s. The incremental value of contour levels is $\mathrm{\Delta [|}S_u^\ast |\textrm{]} = 0.02$.

In range VIII, the cylinder resumes its regular sinusoidal oscillation motion with one dominant frequency, as discussed in figures 4 and 5. The vortex shedding process for $\theta = 140^\circ $ as a representative of range VIII is shown in figure 19, along with the time trace of the cylinder oscillation. The vortex shedding in this range of wire angular locations involves four vortices. When the cylinder is accelerating at the beginning of the cycle in the positive ${y^\ast }$ direction, the first vortex ${V_1}$ is formed. This vortex is moved downstream at position 1. When the cylinder passes the mid-position of the cycle, the second vortex ${V_2}$ with the same vorticity sign as ${V_1}$ is shed, which can be seen at a downstream location later at position 2. During the second half of the cycle, the third vortex ${V_3}$ forms from the smooth side of the cylinder. This vortex is seen at a downstream location at position 3. The fourth vortex ${V_4}$ with the same vorticity sign as ${V_3}$ starts to form slightly after the mid-position of the cycle. In figure 19, ${V_4}$ is seen at a downstream location at position 4 (when the cylinder reaches the end of the cycle). Based on these results, it can be concluded that four vortices are shed in a given oscillation cycle of the wire-fitted cylinder for the wire angular locations of range VIII, and during this shedding process, two vortices of the same vorticity sign (e.g. ${V_1}$ and ${V_2}$) shed one after another. This vortex shedding process is remarkably similar to the 2P vortex shedding mode reported by previous studies for clean cylinders oscillating within the lower synchronization range (Brika & Laneville 1993; Govardhan & Williamson Reference Govardhan and Williamson2000). This similarity between the clean cylinder and the cylinder with the wire attached at the angular locations of range VIII was further verified in the present study by examining the vortex shedding from the clean cylinder undergoing VIV under the same conditions. The results of the oscillating clean cylinder are not presented here for brevity.

Figure 19. Contours of instantaneous normalized vorticity $(\omega D/U)$ for the wire angular position $\theta = 140^\circ $ (from range VIII) along with the time traces of the cylinder displacement. The minimum absolute value and the incremental value of contour levels are: ${[|\omega |D/U]_{min}} = 9$ and $\Delta (\omega D/U) = 1.15$.

3.4.2. Global patterns of velocity spectra

To gain further insight into the unsteady flow characteristics, global representations of velocity spectra have been constructed from the PIV data. This process involved the determination of the autospectra of the streamwise velocity fluctuations at every point in the PIV field of view, detection of the frequency of streamwise velocity fluctuations prevailing over the entire flow field and plotting of the spectral amplitude corresponding to this frequency over the entire near-wake region in the form of contour plots. The contour plots presented in figure 20 show the amplitude of the streamwise velocity spectra, $|{{S_u}(\,f_u^\ast )} |$, at the prevailing frequency of streamwise velocity fluctuations. In this figure, the plots are provided for select wire positions representative of each angular range. The prevailing frequency of velocity fluctuations depends on the wire location $\theta $, and its value is indicated for each $\theta $ in figure 20 in non-dimensional form $f_u^\ast $ (based on the natural frequency of the system, i.e. $f_u^\ast = {f_u}/{f_n}$). Also, to take into account the global effect of the cylinder motion on the near-wake flow field, the mid-position and the oscillation amplitude ${A^\ast }$ are also marked in figure 20 on the images of the cylinder. The wire (not shown in the figure) is positioned on the upper side of the cylinder in these plots.

Figure 20. Contours of the amplitude of the autospectral density corresponding to the streamwise velocity component at the prevailing frequency of velocity fluctuations $|{{S_u}(\,f_u^\ast )} |$ for different wire angular positions that are representative of each wire angular range. The mid-position and the displacement range of the cylinder are also depicted on the image of the cylinder in each plot. Minimum and incremental values are as follows: ${[|{S_u}(\,f_u^\ast )|]_{min}} = 30\;\textrm{mm}\;{\textrm{s}^{ - 1}}$ and $\Delta [|{S_u}(\,f_u^\ast )|] = 2.5\;\textrm{mm}\;{\textrm{s}^{ - 1}}$.

The most salient observation that comes out of the overall inspection of the plots in figure 20 is that, for both wire angular positions of $\theta = 48^\circ $ (from range II) and $\theta = 107^\circ $ (from range VII), the peak spectral amplitudes are relatively low over the whole near-wake region in comparison with the peak spectral amplitudes of the other angular positions. This attenuation is observed not only for the representative $\theta $ given from the angular ranges II and VII, but also for all wire placement angles of these two ranges. As a result, no prominent frequency was detectable for the angular ranges II and VII. The spectral amplitudes in the contour plots of $\theta = 48^\circ $ and $\theta = 107^\circ $ in figure 20 correspond to an arbitrarily selected frequency; however, the same low-amplitude contours existed for all other frequencies at these angles. For the angular ranges II and VII, the non-existence of a dominant frequency corresponding to velocity fluctuations in the wake implies that the strength and coherence of Kármán vortices (indicated by the magnitude of the autospectral density) are drastically attenuated for these wire positions. Notice that this attenuation is more significant in range VII compared with range II because of the observed random switch between the vortex shedding Modes I and II.

For all wire angular positions outside the angular ranges II and VII, a dominant frequency was distinctively identifiable from the spectra of the streamwise velocity component. The peak amplitudes of velocity spectra can ultimately be related to the coherence and strength of periodic vortex shedding. For wire applications that lead to regular sinusoidal motion, such as those within the angular ranges I, V, VI and VIII, very high peak amplitudes are visible in the contour plots of $|{{S_u}(\,f_u^\ast )} |$ in figure 20, indicating highly coherent shedding of vortices. For the wire angular range IV, it was deduced from the time traces of the cylinder motion, given in figure 4, as well as the time–frequency spectrogram of cylinder oscillations, given in figure 6, that the sinusoidal cylinder motion is disturbed intermittently by irregular oscillations. The lower spectral peak observed in velocity fluctuations in figure 20 for the representative case of $\theta = 60^\circ $ from this range compared with the above-mentioned ranges (I, V, VI and VIII) is consistent with the intermittently appearing irregular cylinder motion, pointing out the intimate link between unsteady flow characteristics and the cylinder motion.

The two frequencies dominating the cylinder oscillation motion simultaneously, detected earlier in figure 5, for the wire locations in range III are also the predominant frequencies in the streamwise velocity spectra. Therefore, for the representative case of $\theta = 53^\circ $ in figure 20, two autospectral density contours are provided corresponding to these two frequencies dominating the velocity fluctuations. The peak spectral amplitudes detected in these two plots have comparable levels. However, the locations of these peaks in the near wake are relatively upstream for the lower dominant frequency, reasons of which are not immediately apparent.

3.4.3. Vortex shedding synchronization

Figure 21 depicts the dominant non-dimensional vortex shedding frequency based on the streamwise velocity component $(\,f_u^\ast = {f_u}/{f_n})$ as a function of the wire angular position. For determining the dominant vortex shedding frequency, the global autospectral density of the streamwise velocity signals, obtained from the PIV data, was calculated for a wide range of frequency values, and the frequency with the highest peak magnitude of spectra from the near wake was selected as the dominant frequency. The non-dimensional oscillation frequency of the cylinder $(\,f_d^\ast = {f_d}/{f_n})$ from figure 5 is also shown in this figure for comparison. For the clean cylinder counterpart, it is known that the vortex shedding is synchronized with the oscillation motion as the reduced velocity considered in the present study $({U^\ast } = 10.44)$ is from the lower synchronization range, as discussed earlier in § 3.2. Hence, for the clean cylinder $f_u^\ast = f_d^\ast $. The horizontal line in figure 21 marks the dominant non-dimensional frequency of both the cylinder oscillations and vortex shedding for the clean cylinder.

Figure 21. Variation of the dominant frequency of the oscillation motion $f_d^\ast $ and the dominant frequency of the streamwise velocity fluctuations $f_u^\ast $ with the wire angular position $\theta $ for the oscillating wire-fitted cylinder. The horizontal dashed line marks both the oscillation frequency and the dominant frequency of streamwise velocity signals for the clean cylinder undergoing VIV under the same conditions.

As can be seen from figure 21, similar to the case of the clean cylinder, the vortex shedding of the wire-fitted cylinder is synchronized with the oscillation motion of the cylinder for all angular positions of the wire, except for the angular ranges II $(47^\circ \le \theta < 51^\circ )$ and VII $(105^\circ \le \theta < 109^\circ )$. Hence, the ranges II and VII are hatched in figure 21. It should also be noted that, similar to figure 5, the interpretation of the hatched areas in figure 21 is different for the angular range II and range VII. In range II, the vortex shedding occurs intermittently (as shown in figure 11), and consequently, the cylinder randomly oscillates with sinusoidal and disturbed motions, as seen in figure 4. However, a dominant frequency cannot be selected for the entire period of the cylinder oscillation. On the contrary, in range VII, there is no detectable oscillation motion for the cylinder, and the vortex shedding switches intermittently between Mode I and Mode II (as shown in figure 17). As a result, there is no dominant frequency in range VII.

Another phenomenon revealed in figure 21 is the simultaneous presence of two dominant frequencies throughout the angular range III $(51^\circ \le \theta < 60^\circ )$ not only for the cylinder oscillation motion but also for the vortex shedding process. The oscillation of the wire-fitted cylinder in range III was seen earlier (in figures 4–7) to be dominated by two sinusoidal motions having different frequencies. To demonstrate the existence of dual-frequency domination in velocity fluctuations for this angular range, figure 22 is provided, where the autospectral density of the streamwise u and transverse v components of the velocity, ${S_u}(\,f_u^\ast )$ and ${S_v}(\,f_v^\ast )$, are shown for the wire angular positions of $\theta = 51^\circ $ to $59^\circ $ (angular range III). Each spectrum shown in figure 22 is determined using the velocity signal extracted from the PIV data corresponding to the respective point in the near wake that gives the highest spectral magnitude. As seen in figure 22(a), two distinct peaks are detectable in the spectra of the streamwise velocity component, ${S_u}(\,f_u^\ast )$. The first peak is always the predominant one for all wire angular positions, except for $\theta = 51^\circ $, which is the beginning of range III. On the other hand, figure 22(b) shows that the second peak is the predominant one in the spectra of the transverse velocity component, ${S_v}(\,f_v^\ast )$, for all angular locations in this range. In figure 22, the vertical dashed lines indicate the range of values within which the two dominant frequencies change with the wire angular location, $\theta $. The range of increase in the two dominant frequencies with increasing $\theta $ is identical for flow fluctuations and the displacement of the cylinder, so the vertical dashed lines show the same values in figures 7(c), 22(a) and 22(b).

Figure 22. Frequency spectrum of velocity signals at a point with maximum value of spectral magnitude for wire angular positions of range III ($(51^\circ \le \theta < 60^\circ )$). Existence of two distinct dominant frequencies is depicted for both: (a) the streamwise velocity component u and (b) the transverse velocity component v.

To further address the observed double frequency phenomenon in range III, the time traces of cylinder displacement, ${y^\ast }$, the streamwise velocity component, ${u^\ast }$, and the transverse velocity component, ${v^\ast }$, are provided in figure 23 for $\theta = 53^\circ $ as a representative sample case from the angular range III. The values on the time axis are the same for all plots, and they are presented in non-dimensional form ${t^\ast }$. The velocity signals are extracted from the PIV data at the point that gives the highest spectral magnitude in the near wake, and the spectra given in figure 22 for $\theta = 53^\circ $ correspond to the velocity signals shown in figure 23. The vertical dashed lines in figure 23 indicate one full oscillation cycle of the cylinder, and the dash-dot line represents the start of the second reciprocating motion of the cylinder within the cycle. It was mentioned earlier that the oscillation motion in range III is the superposition of two sinusoids with different frequencies. A careful inspection of the displacement signal ${y^\ast }$ reveals that the first dominant frequency $f_1^\ast $ (the low value) represents one full cycle of the oscillation motion, and the second dominant frequency $f_2^\ast $ (the high value) represents the additional reciprocating motion within the cycle. The two dominant frequencies ($f_1^\ast $ and $f_2^\ast $) are marked in figure 23 for $\theta = 53^\circ $. From the time trace of velocity signals in the same figure, it can be observed that the low- and high-frequency values correspond to the frequency of oscillations in the streamwise and transverse velocity components in the wake, respectively. This observation aligns with the higher spectral magnitude of $f_1^\ast $ seen for ${u^\ast }$ in figure 22(a) and the higher spectral magnitude of $f_2^\ast $ seen for ${v^\ast }$ in figure 22(b). It should be recalled that four vortices are shed per cycle for the angular range III, as discussed in figure 13. Combined with this knowledge, it can be deduced that the first dominant frequency $f_1^\ast $ represents the frequency of the entire vortex shedding process during one oscillation cycle, and the second dominant frequency $f_2^\ast $ can be associated with the frequency of shedding for a pair of vortices.

Figure 23. Time traces of displacement of the cylinder ${y^\ast }$, the streamwise velocity component ${u^\ast }$ and the transverse velocity component ${v^\ast }$ for the wire angular location of $\theta = 53^\circ $ (from range III). The period of the non-dimensionalized time is the same for all plots. One cycle of oscillation motion is marked between the two vertical dashed lines.