4.1. Impact of incoming flow oscillation intensity on VIV response

A first assessment of the impact of flow oscillation intensity $\sigma$ on the cylinder’s response patterns is given with the vibration amplitude $A/D$ across $U^*$ summarised in figure 5(a). For all scenarios with oscillating incoming flows, the reduced velocity $U^*$ was calculated based on the mean flow speed $U_0$ , and $A/D$ was determined from the time series of cylinder displacement with the same approach as discussed in the baseline group section. Results displayed in figure 5 reveal that, despite the velocity oscillations from incoming flow, the cylinder vibration amplitude pattern exhibits a similar trend in comparison with the baseline group, where $A/D$ increases significantly at $U^*\approx 3.8$ and gradually decays within $U^*\gt 6$ . Notably, the growth of $\sigma$ gradually mitigates the maximum cylinder vibration amplitude, which drops from 0.72 $D$ to 0.62 $D$ when $\sigma$ increases from 0 (i.e. baseline group) to 0.3. On the other hand, the presence of incoming flow oscillations results in slight elevation of $A/D$ within the initial branch (i.e. $U^*\leq 3.8$ ). The cylinder vibration amplitude demonstrates minor deviation to the baseline group in the region where $A/D$ exhibits monotonic decay at $U^*\geq 6$ across all investigated $\sigma$ , which indicates the dominant role of unsteady incoming flow in modulating VIV amplitude within the low-reduced-velocity regions.

Figure 6. Collapsed amplitude deviation $(A-A_{\mathrm{QS}})/D$ versus non-dimensional parameter $(U^*-U_p^*)/\sigma$ .

To highlight the non-quasi-steady response of cylinder vibrations in time-varying flow environments, the quasi-steady (hereafter as QS) VIV amplitude (marked as dashed lines) is also included in figure 5(a) for comparison. Here, the QS response $A_{\textit{QS}}$ is constructed by linearly interpolating the baseline steady-flow VIV amplitude as a function of instantaneous reduced velocity $u^*(t) = u(t)/(D f_n)$ , then averaging over one complete flow oscillation cycle. Overall, $A/D$ obtained from the QS response matches the experimental measurements when $U^*\gt 6$ . However, distinctive deviations are observed in regions where $A/D$ is close to its peak values. Specifically, within $3.8\leq U^* \leq 6$ , cylinder vibration amplitude obtained from the QS response is lower than the corresponding experimental results, and this trend is more pronounced with the growth of incoming flow oscillation intensity $\sigma$ . To facilitate the quantification of this deviation, the normalised difference of cylinder vibration amplitude between experimental measurements and QS response, $(A-A_{\textit{QS}})/D$ , is illustrated in figure 5(b). It is evident that regardless of $\sigma$ , $(A-A_{\textit{QS}})/D$ reaches its local maximum at $U^*\approx 4$ , which then reduces with the growth of $U^*$ . When $\sigma$ is small (i.e. $\sigma =0.16$ ), $(A-A_{\textit{QS}})/D$ exhibits a rapid decay which drops to 0 at $U^*\approx 4.8$ . However, at $\sigma =0.3$ , the deviation of cylinder vibration amplitude between experimental measurements and QS response diminishes at a much lower rate. where $(A-A_{\textit{QS}})/D\approx 0$ at $U^*\approx 5.6$ . It is worth noting that the maximum $(A-A_{\textit{QS}})/D$ can reach up to 0.2 at $U^*\approx 4$ , which is equivalent to 50 % of the corresponding cylinder vibration amplitude estimated from the QS response at $\sigma =0.3$ . This highlights the strong non-QS coupling between cylinder motions and fluid loads, which will be analysed in detail in the following sections. To further examine the non-QS amplitude enhancement across different oscillation intensities, we collapse the deviation data $(A-A_{\textit{QS}})/D$ using non-dimensional parameter $(U^*-U^*_p)/\sigma$ , where $U^*_p=4$ represents the reduced velocity with peak amplitude deviation from QS response. Figure 6 presents the amplitude deviation versus this non-dimensional parameter, where $(A-A_{\textit{QS}})/D$ diminishes at $(U^*-U^*_p)/\sigma \approx 5$ across all tested oscillation intensities $\sigma \in [0.16,0.3]$ . Results from figure 6 highlight that the growth of inflow oscillation intensity extends the $U^*$ region within which the VIV amplitude deviates from the QS estimation.

Figure 7. Normalised power spectra $P/\max \{P\}$ of cylinder vibrations under oscillatory flow at $f_w/f_n= 0.18$ under various inflow oscillation intensities $\sigma$ : (a) $\sigma =0.16$ ; (b) $\sigma =0.23$ ; (c) $\sigma =0.3$ . The frequency differences between supplementary components $f_1$ and $f_2$ and the central one $f_0$ are illustrated in the insets.

The impact of incoming flow oscillations on multi-frequency cylinder vibration patterns is revealed via the compensated spectra $P/\max \{P\}$ of cylinder displacements in figure 7 across various $\sigma$ . Notably, instead of a single dominant frequency component from the baseline case shown in figure 3(b), the periodically oscillating incoming velocity results in the presence of three primary cylinder vibration frequency components. Specifically, the central frequency component with the highest energy, denoted as $f_0$ , aligns closely with the dominant one identified in the baseline group. The supplementary frequency components, denoted as $f_1$ and $f_2$ , evolve with an analogous trend to $f_0$ , which exhibits a moderate growth as a function of $U^*$ . To analyse the correlations between these three frequency components, the distribution of $\Delta f_1=f_1-f_0$ and $\Delta f_2=f_2-f_0$ is summarised in the insets of figure 7. Despite the variation of $\sigma$ , both $\Delta f_1$ and $\Delta f_2$ remain consistent across $U^*$ , where their magnitudes are equivalent to the incoming flow oscillation frequency $f_w$ . This phenomenon highlights that velocity fluctuations from incoming flow modulate the VIV response with multi-frequency vibrations through a ‘beat’ mechanism. Additional analysis for the synchronisation between imposed inflow oscillations and VIV amplitude envelopes also evidences strong correlations between these two parameters, with details illustrated in Supplementary Materials § 5. To further quantify the intensity of these secondary vibration frequency components, the Fourier decomposition is implemented, where the cylinder displacement $y(t)$ can then be written in a form with the summation of three harmonic terms

(4.1) \begin{align} \begin{aligned} y(t) &= y_0(t)+y_1(t)+ y_2(t) \\ &=a_0 \sin (2\pi f_0 t+ \phi _0) + a_1 \sin (2\pi f_1 t + \phi _1) + a_2 \sin (2\pi f_2 t + \phi _2) ,\end{aligned} \end{align}

where $a_0$ , $a_1$ and $a_2$ correspond to the vibration amplitude of frequency components at $f_0$ , $f_1$ and $f_2$ , respectively, and $\phi _0$ , $\phi _1$ and $\phi _2$ are the phase shifts of each vibration frequency component with respect to excitation loads. The evolution of normalised $a_1$ and $a_2$ across $U^*$ in figure 8 shows that increasing inflow oscillation intensity $\sigma$ leads to higher $a_1$ and $a_2$ , indicating a more pronounced multi-frequency VIV response. To quantify the relative importance of multi-frequency content, figure 8(c) presents the sideband energy fraction $(a_1^2+a_2^2)/(a_0^2+a_1^2+a_2^2)$ as a function of reduced velocity for the three oscillation intensities. Overall, the sideband energy fraction increases with $\sigma$ across the entire tested range, demonstrating that stronger flow modulation produces more significant energy redistribution from the dominant frequency $f_0$ to the sidebands $f_1$ and $f_2$ . It is worth noting that the stabilisation of sideband fractions at high reduced velocities (i.e. $U^*\geq 6$ ) suggests that the multi-frequency nature of the response is maintained even when the overall VIV amplitude deviations from QS predictions diminish.

Figure 8. Amplitudes of supplementary bands $a_1$ and $a_2$ across $U^*$ with varying inflow oscillation intensity $\sigma$ at fixed frequency ratio $f_w/f_n = 0.18$ . (a) Lower sideband amplitude $a_1/D$ at frequency $f_0 - f_w$ . (b) Upper sideband amplitude $a_2/D$ at frequency $f_0 + f_w$ . (c) Sideband energy fraction $(a_1^2+a_2^2)/(a_0^2+a_1^2+a_2^2)$ .

Investigations for the coupling between fluid loads and the multi-frequency cylinder motions are conducted via analysis of the vibration signal time series. Figure 9 presents the representative time series for normalised cylinder displacements $y/D$ , instantaneous reduced velocity from incoming flow $u^*=u/(f_{\textit{n}}\text{D})$ and force coefficients $C_t$ and $C_v$ . To facilitate the discussion of the phase angle between the fluid loads and the cylinder displacements, the time series of $y/D$ is superimposed in the plots of force coefficients. In general, due to the presence of multi-frequency vibrations, the peak-to-peak cylinder displacements alters across each vibration cycle, and the phase difference between $y/D$ , $C_t$ and $C_v$ exhibits distinctive temporal variations. Specifically, at $U^*=4.05$ , where the difference in VIV amplitude between experimental data and the QS response is most pronounced, the evolution of total fluid load $C_t$ is overall in-phase with $y/D$ at both low ( $\sigma =0.16$ , figure 9 a) and high ( $\sigma =0.3$ , figure 9 b) inflow oscillation intensities. This trend is more clearly reflected in the corresponding Lissajous figures, dominated by the positive-diagonal distributions of $y/D$ and $C_t$ . On the other hand, the vortex load demonstrates anti-phase correlations with cylinder displacement during the majority of the vibration cycles; however, a deviation from this anti-phase correlation is observed when $u^*$ starts to increase during each inflow oscillation cycle. This phenomenon is more distinctive at the high inflow oscillation intensity of $\sigma =0.3$ in figure 9(b), where the correlation between $y/D$ and $C_v$ could intermittently shift to the in-phase counterpart. It is worth stressing that the footprint of in-phase correlation between $y/D$ and $C_v$ is not observed at the same $u^*$ values when $u^*$ is decreasing. This evidences the non-QS VIV response with respect to the instantaneous inflow velocity. When $U^*$ increases to 4.98 (figure 9 c), the total fluid load $C_t$ starts to exhibit anti-phase correlations with respect to $y/D$ when the instantaneous reduced velocity $u^*$ is close to its local maximum. This is also reflected in the presence of anti-diagonal patterns in the Lissajous figure marked by the red lines. Meanwhile, the correlation between $C_v$ and $y/D$ remains anti-phase across all vibration cycles. Further growth of $U^*$ to 5.9 results in a larger portion of anti-phase correlations between $C_t$ and $y/D$ , as shown in figure 9(d).

Figure 9. Time series of cylinder displacement $y/D$ , reduced velocity $u^*$ , total load coefficient $C_t$ and vortex-load coefficient $C_v$ under (a) $U^*=4.05, f_w/f_n=0.18, \sigma =0.16$ ; (b) $U^*=4.05, f_w/f_n=0.18, \sigma =0.30$ ; (c) $U^*=4.98, f_w/f_n=0.18, \sigma =0.30$ ; (d) $U^*=5.90, f_w/f_n=0.18, \sigma =0.30$ . The time series of $C_t$ and $C_v$ are superimposed with background colour, which indicates the absolute value of their phase difference with respect to the cylinder displacement $y/D$ . The associated Lissajous figure of $C_{t}$ versus $y/D$ highlights the temporal variation of phase differences with blue line: $|\phi |\lt$ 90 $^\circ$ , $|\phi _v|\lt$ 90 $^\circ$ ; green line: $|\phi |\lt$ 90 $^\circ$ , $|\phi _v|\gt$ 90 $^\circ$ ; red line: $|\phi |\gt$ 90 $^\circ$ , $|\phi _v|\gt$ 90 $^\circ$ .

To quantify the phase evolution in this critical region $3.8 \lesssim U^* \lesssim 6$ , we conducted additional analysis of duty cycle fractions across different VIV response states, as well as phase distribution statistics. We classified the instantaneous VIV response into three states based on phase angle criteria as: (i) initial: $|\phi _v| \lt 90^\circ$ and $|\phi | \lt 90^\circ$ ; (ii) upper: $|\phi _v| \gt 90^\circ$ and $|\phi | \lt 90^\circ$ ; (iii) lower: $|\phi _v| \gt 90^\circ$ and $|\phi | \gt 90^\circ$ . For each flow oscillation cycle, we computed the fraction of time spent in each state (duty cycle). Violin plot representations display the probability density distribution (mirrored kernel density estimate), with overlaid quartile statistics: the median (thick horizontal line), interquartile range from 25th to 75th percentile (vertical line) and quartile markers (small horizontal ticks).

Figure 10. Duty cycle fractions versus reduced velocity for three oscillation intensities at frequency ratio $f_w/f_n = 0.18$ . (a) Initial state duty fraction. (b) Upper state duty fraction. (c) Lower state duty fraction. The duty fractions quantify the proportion of each flow oscillation cycle spent in each VIV response state.

Figure 10 presents the duty cycle fractions for initial, upper and lower states as functions of reduced velocity for three oscillation intensities ( $\sigma = 0.16$ , 0.23, 0.30). At $U^* \approx 4.05$ (near the initial-to-upper-branch transition), the system predominantly occupies the upper state (duty fraction $\approx 0.8$ ) with a small portion in the initial state, regardless of $\sigma$ . As $U^*$ increases to approximately 5, the upper state duty fraction increases, while the initial state duty fraction decreases to near zero. For $U^*$ reaches 5–6, the lower state duty fraction increases significantly, with a decreasing trend of the upper state. A higher $\sigma = 0.3$ shows a smaller portion in the upper state within $3.8 \lesssim U^* \lesssim 6$ . Figure 11 presents the corresponding violin plots of cycle-averaged phase angles. At $U^* = 4.05$ , the total force phase $\langle |\phi |\rangle _{\textit{cycle}}$ exhibits narrow distributions with range near zero. At $U^* = 4.99$ , the total force phase $\langle |\phi |\rangle _{\textit{cycle}}$ exhibits wider distributions with increased mean, and this trend is amplified with the growth of $\sigma$ . Further increase of $U^*$ to 5.93 results in even higher $\langle |\phi |\rangle _{\textit{cycle}}$ . The vortex-force phase $\langle |\phi _v|\rangle _{\textit{cycle}}$ shows wide distributions centred around 75 $^\circ$ –170 $^\circ$ at $U^* = 4.05$ with a decreasing distribution as $\sigma$ increases. At higher $U^* = 4.99$ and $U^* = 5.93$ , the vortex phase distributions narrow significantly, with $\langle |\phi _v|\rangle _{\textit{cycle}}$ exhibiting particularly tight clustering (interquartile range $\lt$ 10 $^\circ$ ) near 180 $^\circ$ .

Figure 11. Cycle-to-cycle phase distribution at three representative reduced velocities ( $U^* = 4.05$ , $U^* = 4.99$ and $U^* = 5.93$ ) for three oscillation intensities ( $\sigma = 0.16$ , 0.23, 0.30). Top panels: total force phase $\langle |\phi |\rangle _{\textit{cycle}}$ . Bottom panels: vortex-force phase $\langle |\phi _v|\rangle _{\textit{cycle}}$ . Violin plots display the probability density distribution (shaded region) with median (thick horizontal line), interquartile range (vertical line spanning 25th to 75th percentiles) and quartile markers (horizontal ticks at 25th and 75th percentiles).

The temporal evolution of vibration amplitude and its correlation with instantaneous flow conditions are then examined through time series analysis. Figure 12 presents representative time series for normalised cylinder displacement $y/D$ , instantaneous amplitude $A_{\textit{inst}}/D$ extracted via the Hilbert transform, QS amplitude prediction $A_{\textit{QS}}/D$ based on steady-state VIV data and instantaneous reduced velocity $u^*=u/(f_{\textit{n}}\text{D})$ . The instantaneous amplitude $A_{\textit{inst}}/D$ superimposed on the displacement signal reveals the beat-like modulation envelope resulting from the multi-frequency vibration response. The associated Lissajous figures plotting $A_{\textit{inst}}/D$ versus $u^*$ illuminate the hysteresis characteristics of the amplitude response with respect to the instantaneous flow velocity.

Figure 12. Time series of cylinder displacement $y/D$ , reduced velocity $u^*$ , instantaneous amplitude $A_{\textit{inst}}$ and QS amplitude $A_{\textit{QS}}$ under (a) $U^*=4.05, f_w/f_n=0.18, \sigma =0.16$ ; (b) $U^*=4.05, f_w/f_n=0.18, \sigma =0.30$ ; (c) $U^*=4.98, f_w/f_n=0.18, \sigma =0.30$ ; (d) $U^*=5.90, f_w/f_n=0.18, \sigma =0.30$ . The associated Lissajous figure of $A_{\textit{inst}}$ versus $u^*$ highlights the hysteresis effect on instantaneous amplitude.

At $U^*=4.05$ with low flow oscillation intensity ( $\sigma =0.16$ , figure 12 a), the instantaneous amplitude $A_{\textit{inst}}/D$ exhibits moderate temporal variations synchronised with the oscillating flow velocity. The Lissajous figure demonstrates that $A_{\textit{inst}}/D$ does not follow the QS prediction $A_{\textit{QS}}/D$ , with strong hysteresis evident in the trajectory. When the flow oscillation intensity increases to $\sigma =0.3$ at the same $U^*=4.05$ (figure 12 b), both the magnitude of amplitude modulation and the deviation from QS response become more pronounced. Notably, the Lissajous figure reveals stronger hysteretic loops, indicating that the instantaneous amplitude depends not only on the current value of $u^*$ but also on its temporal evolution trend, where the amplitude during the acceleration phase of $u^*$ is smaller than that during the deceleration phase. This hysteresis effect evidences the non-QS nature of the VIV response under unsteady-flow conditions.

As $U^*$ increases to 4.98 (figure 12 c), the hysteresis loops in the Lissajous figure become more complex, with moderate hysteretic loops start generating intersections. Further increase of $U^*$ to 5.9 (figure 12 d) results in the convergence of instantaneous VIV amplitude $A_{\textit{inst}}/D$ to the QS estimations $A_{\textit{QS}}/D$ . Overall, results from figure 12 reveal the deviation between $A_{\textit{inst}}/D$ and $A_{\textit{QS}}/D$ and the pronounced hysteresis effect, in particular at $U^*\approx 4$ , underscoring that the cylinder vibration amplitude cannot be adequately predicted by QS assumptions under these time-varying flow conditions.

Figure 13. The phase-averaged time series of $u^*$ , $\phi$ , $\phi _v$ during one inflow oscillation cycle under (a) $U^*=4.05, f_w/f_n=0.18, \sigma =0.16$ ; (b). $U^*=4.05, f_w/f_n=0.18, \sigma =0.30$ ; (c) $U^*=4.98, f_w/f_n=0.18,{} \sigma =0.30$ ; (d). $U^*=5.90, f_w/f_n=0.18, \sigma =0.30$ . The QS predictions for phases are marked by black dotted lines.

The coupling between cylinder displacements and fluid loads, and its relationship to the instantaneous reduced velocity $u^*$ , are further quantified via the calculation of $\phi$ and $\phi _v$ . Due to the non-sinusoidal vibration patterns containing multiple frequency components in oscillatory inflows, the instantaneous phase angles are determined via a wavelet transform of the vibration signal. Details of this approach are elaborated in Appendix A. Figure 13 demonstrates the phase-averaged evolution of $\phi$ and $\phi _v$ within one incoming flow oscillation cycle for representative cases. To facilitate the discussion, phase angles from QS response ( $\phi _{\textit{QS}}$ and $\phi _{v, \textit{QS}}$ ), which were interpolated from the database in the baseline group (figure 3 d) as a function of instantaneous $u^*$ , are superimposed and marked as dashed lines. Results highlight that, in comparison with the QS response, the evolution of $\phi _v$ exhibits a distinctive hysteresis effect at $U^*=4.05$ (figure 13 a,b). Specifically, with the reduction of instantaneous $u^*$ from its local maximum, the phase angle of the vortex load from the QS response, $\phi _{v, \textit{QS}}$ , abruptly drops from 180 $^\circ$ to 0 $^\circ$ when $u^* \leq 3.8$ . However, $\phi _v$ demonstrates a much slower decaying trend during this time interval. Indeed, under a small $\sigma =0.16$ , $\phi _v$ remains above 120 $^\circ$ across the entire inflow oscillation cycle (figure 13 a). Even under stronger inflow oscillation intensity $\sigma =0.3$ , $\phi _v$ does not drop to 0 $^\circ$ until $t\boldsymbol{\cdot }f_w = 0.7$ (figure 13 b). Notably, the total phase angle $\phi$ remains close to zero throughout the entire inflow oscillation cycle. Under these conditions, during the low $u^*$ intervals (i.e. $u^* \leq 3.8$ ), the coexistence of high $\phi _v$ and small $\phi$ promotes the generation of large fluid load magnitudes $\hat {C_t}$ , resembling those observed within the ‘upper’ branch, as shown in figure 3(c). In contrast, the combination of small $\phi _{v, \textit{QS}}$ and $\phi _{\textit{QS}}$ in the QS response corresponds to the ‘initial’ branch, yielding much lower $\hat {C_t}$ during the same intervals. We therefore conclude that, at $U^* = 4.05$ , the enhanced fluid load magnitudes produced during low $u^*$ intervals account for the higher VIV amplitude $A/D$ in comparison with the QS prediction, as observed in figure 5(a,b).

With the growth of mean reduced velocity $U^*$ from 4 to 6, $\phi _v$ gradually ‘locks’ to 180 $^\circ$ , whereas the total phase angles alter from 0 $^\circ$ to 180 $^\circ$ at high instantaneous $u^*$ intervals (figure 13 c,d). This indicates that the fluid–cylinder coupling intermittently switches between the ‘upper-’ and ‘lower-’branch states within this $U^*$ range. It is worth pointing out that the hysteresis effect observed at $U^*=4.05$ becomes less distinctive at higher $U^*$ , and the phase angles (both total and vortex-load components) obtained from experimental data and the QS response exhibit minor discrepancies when $U^*$ reaches 6. Under these conditions, the VIV patterns during each inflow oscillation cycle are expected to be analogous to those interpolated from the QS estimation, which corresponds to the observation of VIV amplitude as discussed in figure 5.

Figure 14. Phase-averaged amplitude evolution within one flow oscillation cycle and corresponding Lissajous figures. Left panels show temporal evolution of phase-averaged amplitude $A_{\textit{inst}}/D$ (solid line), QS prediction $A_{\textit{QS}}/D$ (dotted line) and instantaneous reduced velocity $u^*$ (blue line) as functions of normalised time $t \boldsymbol{\cdot }f_w$ . Right panels display Lissajous figures plotting $A_{\textit{inst}}/D$ versus $u^*$ with arrows indicating the direction of temporal evolution. Panels show (a) $U^*=4.05$ , $\sigma =0.16$ ; (b) $U^*=4.05$ , $\sigma =0.3$ ; (c) $U^*=4.98$ , $\sigma =0.3$ ; (d) $U^*=5.9$ , $\sigma =0.3$ .

The phase-averaged amplitude response, obtained by ensemble averaging the instantaneous amplitudes at corresponding phases of the flow oscillation cycle, further elucidates the hysteresis phenomena discussed previously. Similar to those observed from instantaneous VIV amplitude in figure 12, at $U^*=4.05$ with low flow oscillation intensity ( $\sigma =0.16$ , figure 14 a), the phase-averaged amplitude $A_{\textit{inst}}/D$ exhibits clear periodic modulation synchronised with the oscillating flow velocity. The Lissajous figure reveals a pronounced hysteresis loop where the ascending phase of $u^*$ (from minimum to maximum) follows a distinct amplitude trajectory compared with the descending phase, forming a counter-clockwise hysteresis loop. Such hysteresis indicates non-QS behaviour, a phenomenon also observed in purely oscillatory VIV where response amplitude and frequency remain path-dependent despite slowly varying flow conditions (Dorogi et al. Reference Dorogi, Baranyi and Konstantinidis2023), where the amplitude during velocity deceleration is consistently higher than the acceleration stage. When the flow oscillation intensity increases to $\sigma =0.3$ at the same $U^*=4.05$ (figure 14 b), the amplitude modulation becomes more pronounced, and the hysteresis loop in the Lissajous figure expands significantly. The deviation between $A_{\textit{inst}}/D$ and $A_{\textit{QS}}/D$ intensifies, with the phase-averaged amplitude remaining elevated during the low $u^*$ intervals compared with QS expectations. This enhanced hysteresis effect indicates that stronger flow oscillations amplify the non-QS response characteristics under this $U^*$ .

As $U^*$ increases to 4.98 (figure 14 c), the phase-averaged amplitude response in Lissajous figure shows a clear intersection in the hysteresis loop at instantaneous $u^*\approx 5.3$ . The trajectory shows reduced deviation from $A_{\textit{QS}}/D$ compared with lower $U^*$ , indicating a gradual transition toward QS behaviour. When $U^*$ reaches 5.9 (figure 14 d), the hysteresis loop becomes narrower and more aligned with the $A_{\textit{QS}}/D$ trajectory, and the hysteresis loop switches to clockwise pattern. The evolution of phase-averaged amplitude patterns from pronounced hysteresis at $U^*=4.05$ under $\sigma =0.3$ to near-QS behaviour at $U^*=5.9$ further evidences that, under combined current–oscillatory inflows, the intensity of non-QS effects is higher under larger $\sigma$ and with $U^*$ closer to the transition from initial to upper branches.

To highlight the effect of oscillation intensity on non-QS amplitude deviation, figure 15 presents the instantaneous amplitude deviation $|A_{\textit{inst}} - A_{\textit{QS}}|/D$ as a function of instantaneous reduced velocity $u^*$ at mean reduced velocity $U^* = 4.05$ . Here, solid lines represent the velocity-decreasing phase (flow deceleration) while dashed lines represent the velocity-increasing phase (flow acceleration). Overall, the results show more distinctive deviations from QS VIV amplitude with increasing $\sigma$ , where the instantaneous $|A_{\textit{inst}} - A_{\textit{QS}}|/D$ reaches up to 0.6 at $\sigma =0.3$ when the inflow velocity phase is close to 270 $^\circ$ (i.e. local minimal of $u^*$ during an flow oscillation cycle). Pronounced hysteresis response between acceleration and deceleration phases is evident across all $\sigma$ values. It is worth noting that $|A_{\textit{inst}} - A_{\textit{QS}}|/D$ exhibits a ‘jump’ at the critical $u^*=4$ (i.e. the velocity when baseline VIV response switches from the ‘initial’ to ‘upper’ branches). Similar phenomenon is observed at other mean reduced velocities.

Figure 15. Instantaneous amplitude deviation $|A_{\textit{inst}} - A_{\textit{QS}}|/D$ versus instantaneous reduced velocity $u^*$ at mean reduced velocity $U^* = 4.05$ for three oscillation intensities ( $\sigma = 0.16, 0.23, 0.30$ ) at fixed frequency ratio $f_w/f_n = 0.18$ . Solid lines represent inflow-velocity-decreasing phase; dashed lines represent inflow-velocity-increasing phase.

Finally, the distribution of time-averaged phase angles $\phi$ and $\phi _v$ across all investigated $U^*$ and $\sigma$ is summarised in figure 16. As discussed in figure 13(a,b), the vortex phase angle $\phi _v$ exhibits temporal variations within each inflow oscillation cycle at $U^*=4.05$ ; this is reflected in the averaged $\phi _v$ locating between 0 $^\circ$ and 180 $^\circ$ . With the growth of $U^*$ , the mean $\phi _v$ converges to 180 $^\circ$ , whereas the total phase angle $\phi$ gradually increases from 0 $^\circ$ to 180 $^\circ$ . As illustrated in figure 13(c,d), the flow velocity variation leads to the coexistence of both ‘upper-’ and ‘lower-’branch states during each incoming flow oscillation cycle. Therefore, instead of an abrupt jump from 0 $^\circ$ to 180 $^\circ$ at $U^*=5.8$ in the baseline group, the mean total phase angle $\phi$ exhibits a moderate increment across $U^*$ , especially for cases with high $\sigma$ . To further examine the influence of multi-frequency VIV response on the mean phase evolution, we conducted detailed frequency band analysis using continuous wavelet transform (CWT, see details of this methodology in Supplementary Material § 8). Figure 17 presents a sensitivity test comparing the time-averaged phase calculated from the full frequency span analysis versus narrow-band analysis using only the $f_0$ band. For both total phase $\phi$ and vortex phase $\phi _v$ , the full-span and $f_0$ -only estimates show in general similar trend across all $U^*$ values. The result confirms that the $f_0$ band dominates the time-averaged phase values. Figure 18 presents the normalised time-averaged weight contributions $w_t$ (total force) and $w_v$ (vortex force) for each frequency band. The $f_0$ band dominates the phase estimation across all conditions, contributing 40 %–60 % of the total weight.

Figure 16. The time-averaged total phase $\phi$ and vortex phase $\phi _v$ between cylinder displacement and fluid forces at inflow oscillation frequency $f_w/f_n = 0.18$ under various oscillation intensities $\sigma$ .

Figure 17. Sensitivity test comparing phase estimates from full frequency span (solid lines, filled markers) versus narrow-band $f_0$ -only analysis (dashed lines, open markers). (a) Total phase $\phi$ . (b) Vortex phase $\phi _v$ .

Figure 18. Normalised time-averaged weight contributions from frequency bands $f_0$ , $f_1$ and $f_2$ for three oscillation intensities. (a) Total force weights $w_t$ . (b) Vortex-force weights $w_v$ .

Figure 19. Time series of phase-averaged instantaneous amplitude, reduced velocity $u^*$ and phase angles $\phi$ and $\phi _v$ under $U^*$ = 4.05, $\sigma$ = 0.16, $f_w/f_n$ = 0.18, the dashed line represents the QS estimation of the left axis quantity, with marked phase location (ad) at different phases of incoming flow velocity; contours underneath show the corresponding phase-average vorticity contours at the peak location of cylinder displacement.

4.2. Wake dynamics at different inflow oscillation intensities

In figure 19, representative normalised phase-averaged vorticity fields according to the incoming flow oscillation cycle, $\omega _zD/u$ , are presented under mean reduced velocity $U^*=4.05$ , $\sigma =0.16$ and $f_w/f_n$ = 0.18. Here, phase-averaged vorticity fields were computed by matching the instantaneous velocity phase with cylinder displacement features. To reveal the coupling mechanism between vortex-shedding modes, cylinder vibration patterns and instantaneous incoming flow velocity, time-varying reduced velocity $u^*$ , and phase-averaged phase angles are included for discussion.

The evolution of the vortex distribution in the cylinder wake flow clearly depicts the transition between various vortex-shedding modes. Here, we compare the vortex-shedding pattern at peak locations of cylinder displacement with the evidence of vortex-shedding pattern transition. Commencing with the phase-averaged fields (a) and (b) when the incoming flow velocity is at the vicinity of its local maximum, the wake flow is characterised by the presence of two vortex pairs within each cylinder vibration cycle, where the split supplementary vortices A2 $_S$ , B2 $_S$ combined with the previously shed main A3 and B3 as a vortex pair. Overall, in comparison with the primary vortex structure within each pair, the strength of the secondary counterpart decays rapidly with the growth of downstream distance. This indicates the ‘2P’ mode within the upper branch (or 2P $_{o}$ mode from the Morse & Williamson (Reference Morse and Williamson2009) work), where the wake features vortex pairs with a weak, oppositely signed secondary vortex structure such as A2 $_S$ and B2 $_S$ . The ‘2P’ vortex-shedding mode remains observable until the instantaneous flow velocity drops to a certain extent, as shown in the phase-averaged field (c) without a clear sign of split vortex C2 $_S$ when the instantaneous $u^*$ reduces to the minimum, which conforms to the slow decay of $\phi _v$ during $t\boldsymbol{\cdot }f_w\in [0,0.5]$ . Notably, from phase-averaged fields (c) to (d), when $u^*$ starts to increase from its local minimum, the near-wake flow right downstream of the cylinder still features a similar distribution of primary vortex-shedding sequence as shown in phase-averaged fields (a) and (b), i.e. the similar spatial distribution of the main vortices C1, C2 and C3 (or D1, D2 and D3) at phase-averaged fields (c) and (d) to vortices A1, A2 and A3 at phase-averaged fields (a). However, the secondary vortex structure is highly suppressed in the near-wake region, where the footprint of split vortex are negligible in (c) and (d). This phenomenon can be further quantified with an objective criterion $|\varGamma ^*_s|/|\varGamma ^*_p|$ , which defines the ratio between secondary-to-primary circulation strength. Here, the non-dimensional circulation was computed as $\varGamma ^* = \varGamma /(uD)$ , where $\varGamma = \int \int \omega _z \, dA$ is the dimensional circulation integrated over the identified vortex core region. The subscripts $s$ and $p$ denote secondary (e.g. A2 $_S$ ) and primary (e.g. A3) vortices, respectively.

Figure 20 presents the phase-resolved circulation ratio for the same case as shown in figure 19, where $|\varGamma ^*_s|/|\varGamma ^*_p|$ exhibits a distinctive reduction from its peak value at $t\boldsymbol{\cdot }f_w\approx 0.25$ , underscoring the suppression of the secondary vortex structure. The results from both the phase-averaged vorticity distribution in figure 19 and the evolution of the circulation ratio in figure 20 indicate a mixed wake mode between the ‘2P’ and ‘2S’ states during the low $u^*$ intervals, where the production and transport of primary vortex structures mirror the phase pattern of the ‘2P’ mode whereas secondary vortex structures are negligible. This phenomenon aligns with the evolution of the vortex phase angle, where the minimum $\phi _v$ falls between $0^\circ$ (i.e. ‘2S’ mode) and $180^\circ$ (i.e. ‘2P’ mode) during the inflow oscillation cycle. Indeed, previous studies have shown that vortex-shedding modes in both active and passive vibrations are strongly influenced by the combined effects of reduced velocity and vibration amplitude (Williamson & Roshko Reference Williamson and Roshko1988; Morse & Williamson Reference Morse and Williamson2009). The present observation of non-QS fluid–structure coupling at low $u^*$ intervals is consistent with these findings: the inertial effects of the vibrating system sustain VIV amplitudes at levels significantly higher than those predicted by QS responses during low instantaneous $u^*$ , such as instant (c). As reported by Morse & Williamson (Reference Morse and Williamson2009) for cylinders under forced vibration, large vibration amplitudes ( $A/D$ ) promote a transition from the ‘2S’ to the 2P $_{o}$ vortex-shedding mode. This transition, in turn, enhances $\hat {C_t}$ and thereby produces VIV amplitudes that exceed those estimated from QS predictions, consistent with the observations in § 4.1.

Figure 20. Circulation ratio $|\varGamma ^*_s|/|\varGamma ^*_p|$ versus flow velocity phase under $U^*$ = 4.05, $\sigma$ = 0.16, $f_w/f_n$ = 0.18. The horizontal dashed line indicates the mixed-mode threshold at 0.1. Values below this threshold indicate suppressed secondary vortex formation.

The vortex distribution patterns at the same $U^*=4.05$ but with a higher inflow oscillation intensity of $\sigma = 0.3$ are shown in figure 21. Similar to the case with $\sigma = 0.16$ in figure 19, the wake is characterised by the ‘2P $_{o}$ ’ vortex-shedding mode from phase-averaged fields (a) to (b) as the inflow velocity decreases from its local maximum, noted by the low strength split vortices A2 $_S$ and B2 $_S$ . The influence of $\sigma$ on vortex production becomes more pronounced when $u^*$ begins to increase. In particular, at phase-averaged fields (c), the near-wake vortex-shedding mode temporarily shifts to the ‘2S’ pattern with the absence of clear vortex split within C2 and C1. This phenomenon does not appear during the inflow oscillation cycle under low $\sigma = 0.16$ , as shown in figure 19. The variation of the vortex-shedding pattern is also consistent with the evolution of $\phi$ and $\phi _v$ , where the phase angles of both total and vortex-induced fluid forces drop to nearly 0 $^\circ$ at $t\boldsymbol{\cdot }f_w \approx 0.7$ .

Figure 21. Time series of phase-averaged instantaneous amplitude, reduced velocity $u^*$ and phase angles $\phi$ and $\phi _v$ under $U^*$ = 4.05, $\sigma$ = 0.3, $f_w/f_n$ = 0.18, the dashed line represents the QS estimation of the left axis quantity, with marked phase location (ac) at different phases of the incoming flow velocity; contours underneath show the corresponding phase-average vorticity contours at the peak location of cylinder displacement.

Figure 22. Time series of phase-averaged instantaneous amplitude, reduced velocity $u^*$ and phase angles $\phi$ and $\phi _v$ under $U^*$ = 4.98, $\sigma$ = 0.3, $f_w/f_n$ = 0.18, the dashed line represents the QS estimation of the left axis quantity, with marked phase location (ad) at different phases of the incoming flow velocity; contours underneath show the corresponding phase-average vorticity contours at the peak location of cylinder displacement.

Figure 23. Circulation ratio $|\varGamma ^*_s|/|\varGamma ^*_p|$ versus flow velocity phase under $U^*$ = 4.98, $\sigma$ = 0.3, $f_w/f_n$ = 0.18.

Figure 24. Cylinder vibration response at fixed inflow oscillation intensity $\sigma = 0.3$ under various frequency ratios $f_w/f_n$ : (a) normalised vibration amplitude $A/D$ ; (b) amplitude difference $A-A_{\textit{QS}}$ between the experimental results and the QS estimations.

Figure 25. Normalised power spectra $P/\max \{P\}$ of cylinder vibrations under oscillatory flow at $\sigma = 0.3$ under various inflow oscillation frequency ratios $f_w/f_n$ : (a) $f_w/f_n=0.27$ ; (b) $f_w/f_n=0.35$ ; (c) $f_w/f_n=0.44$ . The frequency differences between supplementary components $f_1$ and $f_2$ and the central one $f_0$ are illustrated in the insets.

As shown in figure 13, further increasing the mean reduced velocity to $U^*=4.98$ leads to lock-in of the vortex phase angle at $\phi _v \approx 180^\circ$ throughout the entire inflow oscillation cycle, accompanied by a shift of the total phase angle $\phi$ between $0^\circ$ and $180^\circ$ . This behaviour is reflected in the wake vortex distributions presented in figure 22. Across all instantaneous inflow conditions, the wake consistently exhibits two vortex pairs per cylinder vibration cycle, where the secondary vortices A2 $_S$ , B2 $_S$ , C2 $_S$ and D2 $_S$ are paired with the previously shed main vortices A3, B3, C3 and D3. However, the strength of the secondary vortex structures undergoes pronounced temporal variations. At the phase-averaged field (a) when $u^*$ is near its local maximum, the secondary vortex A2 $_S$ that convects into the wake is comparable in strength to the previously shed primary counterpart A3. As $u^*$ decreases, the relative strength of secondary vortex diminishes significantly, as illustrated at phase-averaged fields (c) and (d). This effect is quantified by the circulation ratio $|\varGamma^*_s|/|\varGamma^*_p|$ between secondary and primary vortices (i.e. the circulation ratio between vortices A2 $_S$ , B2 $_S$ , C2 $_S$ , D2 $_S$ and vortices A3, B3, C3, D3 shed from the previous cycle) as shown in figure 23, which decreases from maximum near the velocity peak to minimal when the inflow velocity also approaches the local minimum, marking the transition from the ‘2P’ to the ‘2P $_{o}$ ’ shedding mode. The temporal evolution of vortex-shedding modes is consistent with the phase dynamics of the fluid forces: the ‘2P’ mode dominates when the associated $\phi$ and $\phi _v$ approach $180^\circ$ (the ‘lower’ branch), whereas the wake transitions to ‘2P $_{o}$ ’ mode when $\phi \approx 0^\circ$ and $\phi _v \approx 180^\circ$ .

4.3. Impact of incoming flow oscillation frequency on VIV response

In this section, we explore VIV responses of the circular cylinder under periodically oscillating flow conditions with a fixed oscillation intensity $\sigma = 0.3$ and various frequencies $f_w/f_n= [0.27,0.35,0.44]$ (i.e. group 3 from table 1). The cylinder vibration amplitudes $A/D$ superimposed with values interpolated from the QS response are summarised in figure 24. To facilitate the discussion, experimental results from case 2.3 (i.e. $\sigma = 0.3$ and $f_w/f_n=0.18$ ) under the same $U^*$ ranges are also included. Overall, similar to the results presented in figure 5 with varying flow oscillation intensities, the VIV amplitudes surpass those predicted by the QS estimation, yet remain lower than those recorded from the baseline group under steady flows within $3.8\leq U^*\leq 6$ . This phenomenon is more clearly reflected in the deviation of the cylinder vibration amplitude between experimental data and the QS response in figure 24(b), where $(A-A_{\textit{QS}})/D$ reaches the maximum at $U^*\approx 4$ and drops to 0 at $U^*\approx 5.6$ , regardless of $f_w$ .

To further reveal the impact of incoming flow oscillation frequency on the multi-frequency cylinder vibration response, the compensated spectra of cylinder displacement across various $f_w$ are illustrated in figure 25. Similar to those discussed in figure 7, the dominant frequency component $f_0$ remains consistent with that in steady-flow scenarios. This indicates that both incoming flow oscillation intensity and frequency have minor influence on the dominant VIV frequency within the investigated parameter space. On the other hand, the distribution of supplementary vibration frequency components $f_1$ and $f_2$ follows well the ‘beat’ mechanism with $f_1\approx f_0-f_w$ and $f_2\approx f_0+f_w$ . It is worth pointing out that overall, with the growth of $f_w$ , the energy associated with these supplementary vibration frequency components progressively weakens. This phenomenon is quantified by the distribution of $a_1/D$ and $a_2/D$ in figure 26, where higher frequency ratios $f_w/f_n$ produce reduced sideband amplitudes across the entire tested range. To quantify the relative importance of multi-frequency content as a function of frequency ratios, figure 26(c) presents the sideband energy fraction $(a_1^2+a_2^2)/(a_0^2+a_1^2+a_2^2)$ at different frequency ratios under constant oscillation intensity $\sigma = 0.30$ , where the sideband energy fraction decreases systematically with increasing $f_w/f_n$ . We will further explore this multi-frequency fluid–structure coupling mechanism via theoretical interpretations in § 5.

Figure 26. Amplitudes of supplementary bands $a_1$ and $a_2$ across $U^*$ with varying inflow oscillation frequency ratios $f_w/f_n$ at constant oscillation intensity $\sigma = 0.30$ . (a) Lower sideband amplitude $a_1/D$ at frequency $f_0 - f_w$ . (b) Upper sideband amplitude $a_2/D$ at frequency $f_0 + f_w$ . (c) Sideband energy fraction $(a_1^2+a_2^2)/{}(a_0^2+a_1^2+a_2^2)$ .

The associated phase-averaged evolution of $\phi$ and $\phi _v$ within one incoming flow oscillation cycle is shown in figure 27. As illustrated in figure 24, cylinder vibrations exhibit the strongest deviations from the QS response at $U^*\approx 4$ . Therefore, our discussion will focus on this critical $U^*$ to reveal the impact of $f_w$ on the VIV response. Overall, regardless of $f_w$ , the phase-averaged $\phi$ between total fluid load and cylinder displacement remains close to 0 $^\circ$ and exhibits minor deviations from the QS response at this $U^*$ . On the other hand, the variation of $f_w$ highly modulates the coupling between vortex fluid load and cylinder motions. Specifically, under low $f_w/f_n$ = 0.18 and 0.27, $\phi _v$ demonstrates distinctive temporal variations (figure 27 a,b) during one inflow oscillation cycle, indicating the intermittent shifting between in-phase and anti-phase correlations between vortex load and cylinder vibrations. However, the footprint of the in-phase correlation (i.e. $\phi _v\approx 0^\circ$ ) gradually disappears with the growth of $f_w$ as shown in figure 27(c,d), and $\phi _v$ is nearly ‘locked’ to 180 $^\circ$ when $f_w/f_n$ reaches 0.44 (figure 27 d). In such a scenario (i.e. $\phi \approx 0^\circ$ and $\phi _v\approx 180^\circ$ ), the wake–cylinder interaction pattern is ‘locked’ within the ‘upper’ branch during the entire inflow oscillation cycle, despite the estimation from the QS response indicating the presence of the ‘initial’ branch when $u^*\leq 3.8$ .

Figure 27. The phase-averaged time series of $u^*$ , $\phi$ , $\phi _v$ during one inflow oscillation cycle under $U^*=4.05$ with $\sigma = 0.3$ at (a) $f_w/f_n=0.18$ ; (b) $f_w/f_n=0.27$ ; (c) $f_w/f_n=0.35$ ; (d) $f_w/f_n=0.44$ . The QS predictions for phases are marked by black dotted lines.

The influence of wave frequency on the phase-averaged amplitude response is further examined at constant $U^*=4.05$ where the measured amplitudes exhibit the most distinctive variation in comparison with the QS estimations (figure 28). Overall, despite the variation of $f_w$ , the evolution of $A_{\textit{inst}}/D$ exhibits a similar trend. Specifically, $A_{\textit{inst}}/D$ remains at 0.58 $\pm$ 0.1 across the inflow oscillation cycle, with the Lissajous figure displaying a well-defined elliptical hysteresis loop. The deviation from the QS prediction $A_{\textit{QS}}/D$ is evident throughout the oscillation cycle, indicating significant non-QS effects at this $U^*$ . The counter-clockwise trajectory in the Lissajous figure confirms that the amplitude response lags behind the instantaneous changes in $u^*$ . The consistency of hysteretic behaviour across the examined frequency range indicates that, within this parameter space, the non-QS nature of the VIV amplitude response is relatively insensitive to variations in wave frequency.

Figure 28. Phase-averaged amplitude evolution within one flow oscillation cycle at varying wave frequencies. Left panels show temporal evolution of phase-averaged amplitude $A_{\textit{inst}}/D$ (solid line), QS prediction $A_{\textit{QS}}/D$ (dotted line) and instantaneous reduced velocity $u^*$ (blue line) as functions of normalised time $t \boldsymbol{\cdot }f_w$ . Right panels display Lissajous figures plotting $A_{\textit{inst}}/D$ versus $u^*$ with arrows indicating the direction of temporal evolution. Panels show (a) $U^*=4.05$ , $f_w/f_n=0.18$ , $\sigma =0.3$ ; (b) $U^*=4.05$ , $f_w/f_n=0.27$ , $\sigma =0.3$ ; (c) $U^*=4.05$ , $f_w/f_n=0.35$ , $\sigma =0.3$ ; (d) $U^*=4.05$ , $f_w/f_n=0.44$ , $\sigma =0.3$ .

Similar to the analysis in figure 15, to reflect the effect of oscillation frequency on non-QS amplitude deviation, figure 29 presents the instantaneous amplitude deviation $|A_{\textit{inst}} - A_{\textit{QS}}|/D$ as a function of instantaneous reduced velocity $u^*$ at mean $U^* = 4.05$ for four frequency ratios at fixed oscillation intensity $\sigma = 0.30$ . The results show a weak dependence of $|A_{\textit{inst}} - A_{\textit{QS}}|/D$ on the frequency ratio compared with the strong dependence on the oscillation intensity observed in figure 15. Across all tested frequency ratios ( $f_w/f_n = 0.18$ to $0.44$ ), the highest deviations consistently occur in the low inflow speed region during the flow velocity deceleration phase. The overall magnitude and distribution of deviations remain relatively similar across different $f_w/f_n$ , further indicating that the non-QS VIV amplitude response is primarily controlled by inflow oscillation intensity rather than oscillation frequency within the tested parameter range.

Figure 29. Instantaneous amplitude deviation $|A_{\textit{inst}} - A_{\textit{QS}}|/D$ versus instantaneous reduced velocity $u^*$ at mean reduced velocity $U^* = 4.05$ for four frequency ratios ( $f_w/f_n = 0.18, 0.27, 0.35, 0.44$ ) at fixed oscillation intensity $\sigma = 0.30$ . Solid lines represent inflow-velocity-decreasing phase; dashed lines represent inflow-velocity-increasing phase.

In summary, our experimental measurements for the VIV response of the circular cylinder in oscillatory incoming flows across various oscillation intensities and frequencies highlight the presence of multi-frequency cylinder vibration patterns and the distinctive non-QS response with respect to instantaneous inflow velocities. This phenomenon is mostly distinctive within $3.8\leq U^*\leq 6$ , which corresponds to the ‘upper’ branch of VIV under steady-flow conditions.

4.4. Wake dynamics at different inflow oscillation frequencies

The impacts of inflow oscillation frequencies on vortex production and transport in the wake flow are firstly elaborated in figure 30 under the critical $U^*=4.05$ at $\sigma =0.3$ with $f_w/f_n=0.27$ . Overall, the wake flow features two vortex pairs per cylinder vibration cycle when $u^*$ is close to its local maximum at phase-averaged field (a). Similar pattern is observed at phase-averaged field (b) with a lower vorticity strength due to the smaller instantaneous $u^*$ . This is consistent with the occurrence of a small total fluid load phase $\phi \approx 0^\circ$ and a large vortex-load phase $\phi _v\approx 180^\circ$ during this phase-averaged field. As discussed in figure 27 with phase-averaged $\phi$ and $\phi _v$ , the growth of $f_w$ mitigates the in-phase correlation between vortex load and cylinder displacements during the $u^*$ growth intervals. Therefore, in contrast to scenarios at a lower $f_w/f_n=0.18$ in figure 21, the typical ‘2S’ vortex-shedding mode is not observed within these intervals. Notably, the minimum $\phi _v$ reaches approximately 120 $^\circ$ across the flow oscillation cycle at $f_w/f_n=0.27$ . Therefore, similar to the inflow condition with $U^*$ = 4.05, $f_w/f_n=0.18$ and $\sigma =0.16$ discussed in figure 19, the wake flow features a mixed status between ‘2P’ and ‘2S’ modes during the $u^*$ growth intervals, where the distribution of primary vortex structures C1(D1) and C3(D3) resembles A1 and A3 at phase-averaged field (a) but with suppressed vortex split.

Figure 30. Time series of phase-averaged instantaneous amplitude, reduced velocity $u^*$ and phase angles $\phi$ and $\phi _v$ under $U^*$ = 4.05, $\sigma$ = 0.3, $f_w/f_n$ = 0.27, the dashed line represents the QS estimation of the left axis quantity, with marked phase location (ad) at different phases of the incoming flow velocity; contours underneath show the corresponding phase-average vorticity contours at the peak location of cylinder displacement.

Figure 31. Time series of phase-averaged instantaneous amplitude, reduced velocity $u^*$ and and phase angles $\phi$ , $\phi _v$ under $U^*$ = 4.05, $\sigma$ = 0.3, $f_w/f_n$ = 0.44; the dashed line represents the QS estimation of the left axis quantity, with marked phase location (a) and (b) at different phases of the incoming flow velocity; (i) distribution of normalised vorticity contours; (ii) distribution of normalised strain rate, the vorticity isolines are adopted from (i).

Finally, when the frequency ratio increases to $f_w/f_n=0.44$ , the vortex-load phase becomes fully locked at $180^\circ$ throughout the entire inflow oscillation cycle. Under these conditions, the near wake immediately downstream of the cylinder is characterised by two vortex pairs across the full cycle, as illustrated in figure 31. Nevertheless, the patterns of vortex transport exhibit distinct differences between the $u^*$ reduction and growth intervals. Specifically, as $u^*$ decreases from its local maximum to minimum (phase-averaged field (a-i)), the vortex structure (A2 $_S$ , A3) remains relatively intact within $0.5 \leq x/D \leq 4$ . In contrast, during the $u^*$ growth interval (phase-averaged field (b-i)), the shed vortices rapidly disintegrate, and the wake is dominated by the coexistence of small, oppositely signed vortices that mix with one another (i.e. the vortex structure B1 to B3), thereby disrupting the coherence of the wake structure. This phenomenon can be attributed to variations in the wake strain rate, as shown in the corresponding strain rate contours in figures 31(a-ii) and (b-ii). The results indicate that, compared with the $u^*$ reduction intervals such as phase-averaged field (a), the rapid increase of inflow velocity produces a more dispersed distribution of intense strain rate region in the near-wake field. As pointed out by Govardhan & Williamson (Reference Govardhan and Williamson2000), these high strain rate regions lead to the split of vortex structures, resulting in a more chaotic wake dynamics during inflow acceleration intervals.