2.1.1. Flow regimes

The steady flow around a surface-piercing hydrofoil can be categorized as FW, partially ventilated (PV) or FV, distinguished by the extent and the stability of a ventilated gas cavity, as defined by Breslin & Skalak (Reference Breslin and Skalak1959), Harwood et al. (Reference Harwood, Young and Ceccio2016a) and Young et al. (Reference Young, Harwood, Montero, Ward and Ceccio2017).

2.1.2. Formation mechanisms of ventilation

The formation of a ventilated cavity can be deconstructed into two stages: inception and stabilization. Inception marks the transition from a FW flow to a PV flow, where the gas cavity begins to grow. Stabilization occurs as the cavity's size and the streamline kinematics reach a critical stability condition, at which point it is categorized as FV flow. Ventilation inception mechanisms rely both on the existence of a low-pressure region, as well as a path that can continuously entrain and transport gas to this region (Rothblum Reference Rothblum1977). The various mechanisms by which flow separation and/or air pathways produce ventilation formation have been extensively categorized (Swales et al. Reference Swales, Wright, McGregor and Rothblum1974; Young et al. Reference Young, Harwood, Montero, Ward and Ceccio2017), and a few dominant mechanisms are summarized below.

1. (a) Stall-induced ventilation occurs at high angles of attack as a result of massive flow separation from a foil's suction surface. Air is entrained either through the gradual depression of the free surface or through the low-energy pathways present in the vortical shear layer produced by stall.

2. (b) Tip-vortex-induced ventilation is a self-initiated mechanism for ventilation, caused by the aeration of a tip vortex. The large angular velocities of tip vortices cause low core pressures that will ingest air from the free surface at the distal end of the vortex. The persistence of tip vortices means that they may ingest air even very far downstream from the foil. When vortex core cavitation occurs, the vortex trajectory is pushed toward the free surface by buoyancy as well.

3. (c) Tail ventilation occurs when low pressures produce a strong downwards acceleration of the free surface. This in turn produces unstable growth of small perturbations (ripples, vortices, etc) that can draw air into ventilation-prone areas. These so-called Taylor instabilities (Taylor Reference Taylor1950) can be especially important when they occur on a thin sheet of liquid water separating the free surface from a vaporous cavity (Rothblum Reference Rothblum1977).

2.1.3. Elimination mechanisms of ventilation

The elimination of ventilated cavities encompasses the washout and rewetting events, respectively describing the transition from FV to PV and from PV to FW flows. Elimination tends to be driven by re-entrant jet instability at low speeds and moderate angles of attack (Harwood et al. Reference Harwood, Young and Ceccio2016a), turbulent reattachment at lower angles of attack and/or higher speeds (Young et al. Reference Young, Harwood, Montero, Ward and Ceccio2017) or cavity choking at high speeds, where a small airway feeding a larger submerged cavity collapses under the very low pressure produced by the constrained airflow (Wadlin Reference Wadlin1959; Elata Reference Elata1967).

2.1.4. Hysteresis response of ventilation

The hysteresis response of a surface-piercing foil is visible on both the lift and moment curves when plotted against the angle of attack curve or the Froude number. It has been shown that FV flow returns to the FW flow at a lower angle of attack and speed than the values at which ventilation was initiated, describing a hysteresis loop. The control of a ventilated foil is severely compromised as a result of this hysteresis, and the flow can assume several alternate, locally stable flow regimes (with widely varying hydrodynamic forces and moments) across a large range of speeds and angles of attack (Breslin & Skalak Reference Breslin and Skalak1959; Harwood et al. Reference Harwood, Young and Ceccio2016a). As a result, large changes in the attack angle, speed or both may be necessary to ensure that flow is unambiguously wetted or ventilated.

4.3.1. Output-only modal analysis

Parameter estimation was used to extract significant operational modal properties from the output data. This class of data analysis is often referred to as output-only or operational modal analysis. These methods were developed for the study of structural modes and responses when there is no input available, and it is used here for the analysis of the operational modal characteristics across a range of flow conditions. The covariance-driven stochastic subspace identification (SSI-COV) algorithm (Aoki Reference Aoki1987; Peeters & De Roeck Reference Peeters and De Roeck1999) was used for computational efficiency. The flow chart of the SSI-COV algorithm used to compute the modal eigenfrequencies (modes) and mode shape is shown in figure 36. A more detailed description of the algorithmic procedure is given in § A.1 in the Appendix A. As with any output-only method, a few caveats apply. The modes from an output-only analysis should be treated carefully. Peaks in the excitation spectra – such as those caused by periodic hydrodynamic phenomena – will cause forced-vibration responses that may be interpreted as structural resonance. Stochastic subspace identification also produces complex-valued modes, wherein node lines change in time. In this work, each complex mode shape was rotated to maximize the Euclidean norm of its real part, which was shown by Ahmadian, Gladwell & Ismail (Reference Ahmadian, Gladwell and Ismail1995) to maximize the similarity between a complex mode and its associated (real) normal mode shape. The real part was then used for plotting, discarding phase information. The complex modes can be viewed as animated movies in the online supplementary materials available at http://doi.org/10.1017/jfm.2023.127. The associated mode shapes should also be more correctly referred to as operating deflection shapes (ODS). In the following results and discussion, the term mode shape will be used in association with a structural resonance, while ODS will be used to describe the shape of the forced-vibration response of the hydrofoil. Finally, the damping estimates from SSI are also influenced by the shape of the input spectrum, so the reported damping ratios should be interpreted qualitatively, rather than quantitatively.

4.3.2. Spectral proper orthogonal decomposition of video

Spectral proper orthogonal decomposition (SPOD) was used to identify energy ranked and spatio-temporally orthogonal modes (Schmidt et al. Reference Schmidt, Towne, Rigas, Colonius and Bres2018; Towne, Schmidt & Colonius Reference Towne, Schmidt and Colonius2018a) from high-speed video records for each run. Smith et al. (Reference Smith, Venning, Pearce, Young and Brandner2020a,Reference Smith, Venning, Pearce, Young and Brandnerb) used SPOD to excellent effect to explore the patterning of cavitation shedding on a flexible hydrofoil. In this work, SPOD was performed on the video recordings from the underwater high-speed camera imaging the foil's suction surface, with durations ranging from 25 to 45 s. Each video was cropped to a region of interest containing the suction side of the hydrofoil, the free surface and the visible portion of the wake. The MATLAB SPOD toolbox published by Towne et al. (Reference Towne, Schmidt and Colonius2018a), Schmidt et al. (Reference Schmidt, Towne, Rigas, Colonius and Bres2018) was used for the decomposition. To further economize calculations, frames were downsampled to a rate of 166.67 Hz. Windowing was specified with 2048 frames with a 75 % overlap, and only the single most energetic mode at each frequency was selected for plotting in the results to follow.

4.3.3. Co-analysis procedure

A co-analysis framework proposed by Di Napoli (Reference Di Napoli2022) was used to illustrate the interactions between ambient hydrodynamic modes and structural modes. The co-analysis workflow is depicted in figure 5. For a given run, the SPOD spectrum was pre-computed for the selected region of interest of the video. Stochastic subspace identification was then performed on a subset of ten bending, ten torsion and single yawing-moment time series, with a model order interactively specified by the user, producing a set of candidate poles representing both resonant modes and flow-induced vibration. The SPOD videographic modes were then extracted at only those frequencies most closely matching the imaginary component of each pole fitted by SSI. These videographic modes indicate the presence of any spatially coherent flow patterns that may be associated with – and occurring at the same frequency as – the structural vibration, thus disambiguating the candidate pole as a system resonance, a forced vibration or a spurious artifact. In the following results, distinctions will be made between the hydrodynamic modes and the structural modes or ODS. Additional details about the SPOD analysis can be found in § A.1.4 in the Appendix A.

Figure 5. Workflow for SSI-SPOD co-analysis. Here SSI is used to estimate system poles from SS bending and twisting measurements. Those poles are used to specify the frequencies at which synchronous videographic SPOD modes are extracted. Both resonant and forced-vibration responses are thus analysed in both solid and fluid domains, allowing the causality of the hydro-structural responses to be explored.

6.1.1. $\alpha = 5^\circ, AR_h = 1, F_{nh} = 1.5$

Figure 7 shows photos (taken from above and below the water surface) of two FW runs: run 1001 in CW in (a) and run 1601 in waves in (b). Both runs are identical except for the presence of regular waves (with wave amplitude $A_w$ of 0.05 m and wave period $T_w$ of 1.5 s) in run 1601. The corresponding time histories of the hydrodynamic load coefficients and tip deformations for both runs are compared in figure 8. Figure 9 compares the PSDs of the fluctuating hydrodynamic load coefficients and tip deformations.

Figure 7. Above water view (top) and underwater view (bottom) of the suction side of the foil for a CW run (a) and a run in waves with $T_w=1.5$ s, $A_w=0.05$ m (b) at $\alpha = 5^\circ$, $AR_h=1$, $F_n=1.5$, $P = P_{atm}$. Two snapshots are shown for the run in waves, the first showing a wave trough and the second a wave crest at the location of the foil's leading edge. Both runs are FW. The free surface is depressed near the foil TE due to the suction created by the foil. An aerated triangular base cavity forms in the separated flow behind the TE, the length of which decreases with increasing depth (and hydrostatic pressure). Aerated von Kármán vortices are also visible in the underwater photos as bubbly striations behind the aerated base cavity. (a) Calm Water, Run 1001, $\alpha =5^\circ$. (b) Waves, Run 1601, $\alpha =5^\circ$.

Figure 8. Time histories of the hydrodynamic coefficients on the top and normalized tip deflections on the bottom for a CW run (a) and a run in waves with $T_w=1.5$ s, $A_w=0.05$ m (b) at $\alpha =5^\circ$, $AR_h=1$, $F_{nh}=1.5$, $P_t = P_{atm}$. Both runs are FW ($\textrm {iFV} = 0$). The horizontal black dashed lines indicate the average of the values in the steady-speed region. The mean values are practically the same in CW and in waves, but waves introduce oscillations at the encounter frequency.

Figure 9. Power spectral density of the fluctuating hydrodynamic load coefficients on the top and tip deformations on the bottom for a CW run (a) and a wave ($T_w=1.5$ s, $A_w=0.05$ m) run (b) at $\alpha = 5^\circ$, $AR_h=1$, $F_{nh}=1.5$, $P = P_{atm}$. The carriage and foil modal frequencies ($\,f_{car},f_{1-4}$) are indicated by the vertical green and blue dashed-dotted lines, respectively. The wave encounter frequency ($\,f_e$) and the vortex shedding frequency ($\,f_{vs}$) are indicated by the vertical red and orange dashed lines, respectively. The frequency response spectra are very similar between the CW and wave cases, except for the addition of a dominant peak at $\,f_e$ and the spreading of the peak near $\,f_3$ for the wave run.

At a low angle of attack, both CW and wave cases are FW, as shown in figure 7 with $\alpha = 5^\circ$. Regular waves introduce periodic oscillations in forces and deflections about the mean but do not alter those means. The frequency response of both the hydrodynamic load coefficients and tip deformations show peaks at the foil modal frequencies ($\,f_1,f_2,f_3$), the carriage modal frequency ($\,f_{car}$) and the vortex shedding frequency ($\,f_{vs}$, as defined in (4.2)). The presence of waves introduces a peak at the wave encountered frequency ($\,f_e$, as defined in (4.1)), but the vortex shedding frequency ($\,f_{vs}$) and the foil modal frequencies remain the same. One effect of the waves is apparent in the shape of the spectra near the third foil resonance ($\,f_3$) in figure 9, where the presence of waves causes the peak to spread out into a less distinct plateau. The small, periodic changes in immersion depth produced by the 0.05 m amplitude waves are expected (based upon the results of Harwood et al. (Reference Harwood, Felli, Falchi, Ceccio and Young2019) and those shown in figure 3) to alter the natural frequency of mode 3 by approximately $-$2.9 Hz to 0.85 Hz at the wave peaks and troughs, respectively. This spread of approximately 4 Hz closely matches the shape of the spectrum in figure 9. Peaks for modes 1 and 2 are either less strongly affected by small changes in immersion (mode 1) or are masked by the high damping and low peak power (mode 2).

6.1.2. $\alpha = 15^\circ, AR_h = 1, F_{nh} = 1.5$

The test data show that waves had no effect upon the flow regime when it is operating sufficiently far away (at least 2 to 3 degrees) from the ventilation boundary, which is at $\alpha \approx 15^\circ$ for $AR_h = 1$ and $F_{nh} = 1.5$ (as shown in figure 6). However, very near the ventilation boundary, the presence of shallow, regular waves tended to delay the transition from FW to FV flows, illustrated by comparing a CW and a wave run at $\alpha = 15^\circ$, $AR_h = 1$, $F_{nh} = 1.5$ and $P_t = P_{atm}$. Photographs from above and below the water surface are shown in figure 10, and the corresponding time histories and PSDs of the load coefficients and tip deformations are shown in figures 11 and 12.

Figure 10. Above water view (top) of the surface pattern and underwater view (bottom) of the suction side of the foil for a CW run (a) and a wave ($T_w=1.5$ s, $A_w=0.05$ m) run (b), both at $\alpha = 15^\circ$, $AR_h = 1$, $F_{nh} = 1.5$, $P_t = P_{atm}$. Two snapshots are shown for the wave run, where the first one corresponds to the wave trough and the second one corresponds to the wave crest. The CW run on the left is FV, as evident by the glassy gaseous cavity that reached all the way to the tip at the foil leading edge. The wave run on the right is FW, illustrating that the presence of regular long waves tends to delay FW-to-FV transition. The spacing between the aerated von Kármán vortices in the wake for run 4801 is half of that of run 5001 due to subharmonic lock-in.

Figure 11. Time histories of lift and moment coefficients ($C_L, C_M$) for a CW run (a) and a wave ($T_w=1.5$ s, $A_w=0.05$ m) run (b) at $\alpha = 15^\circ$, $AR_h = 1$, $F_n = 1.5$, $P = P_{atm}$. The thin red line indicates $F_{ni}=U_i/\sqrt {gh}$ defined based on the instantaneous carriage velocity $U_i$. The thin black dashed line, the thick magenta solid line and the thin blue solid line indicate the raw, moving time average and fluctuating values, respectively. While the mean lift coefficients in the steady-speed region are approximately the same for the CW and wave runs, the mean moment coefficient is significantly lower for the FV run on the left (CW) than that of the FW run on the right (waves).

Figure 12. Time-frequency spectra of the tip twist deformation ($\theta$) for a CW run (a) and run in waves ($T_w=1.5$ s; $A_w=0.05$ m) run (b) at $\alpha = 15^\circ$, $AR_h = 1$, $F_{nh} = 1.5$, $P_t = P_{atm}$. The top panel shows the time history of ($\theta$) (blue solid line) and the instantaneous Froude number ($F_{ni}=U_i/\sqrt {gh}$) (red dashed line). The lower left panel indicates the power spectra (PS) and the lower right panel indicates the time-frequency spectra of $\theta$. The cyan-coloured triangles on the $y$ axis of the lower two panels mark the first three modal frequencies ($\,f_1, f_2, f_3$) of the hydrofoil. The modal frequencies are higher in the FV flow (left case) than in the FW flow (right case) because of a reduction in added mass when a majority of the suction side is covered by the aerated cavity. The carriage modal frequency ($\,f_{car}$) is indicated by the green crosses. The vortex shedding frequency ($\,f_{vs}$) is indicated by the magenta dashed line and the wave encounter frequency ($\,f_{e}$) is indicated by the red dotted line in the lower panels. Compared with the FW run 5001, there is an absence of a peak at $\,f_e$, and the intensity of the peaks at $\,f_{vs}$ and $\,f_3$ are higher due to subharmonic lock-in for the FV run 4801 in CW.

At $\alpha = 15^\circ$, $AR_h = 1$, $F_{nh} = 1.5$ and $P_t = P_{atm}$, the CW case (run 4801) is FV while the wave case (run 5001) is FW. These two runs have identical run conditions except for the presence of small-amplitude waves in run 5001, suggesting that long waves delay transition to ventilation. We hypothesize that the periodic wave-induced pressure and velocity fluctuations produce similar transience in the separated flow on the suction surface. Ventilation inception necessitates a stable region of low pressure and low momentum as well as a continuous path that connects this low-energy region to the free surface; waves interrupt the stability of the low-energy region, delaying ventilation. As shown in figure 11, while the mean $C_L$ and $\delta$ remain approximately the same, the mean $C_M$ and $\theta$ are both lower in the steady-speed region in the FV flow for the CW case compared with the FW flow for the wave case. As described earlier, ventilation causes the centre of pressure to move toward the mid-chord, producing a measurable reduction in $C_M$ because a majority of the suction surface is enveloped in a constant pressure cavity. Note that the mean of the moving time average (represented by the thick magenta solid line in figure 11) values in the steady-speed region corresponds to one of the data points shown in figure 6. Since the foil exhibits a linear elastic response, $\delta$ correlates with $C_L$ and $\theta$ correlates with $C_M$. The sensitivity of both $C_M$ and $\theta$ to the centre of pressure location suggests that they are better indicators of separation and ventilation than $C_L$ and $\delta$.

Figure 12 shows the time-frequency spectra of the tip twist deformation ($\theta$) for the FV flow in CW (panel a) and the FW flow in waves (panel b). In figure 12 the cyan-coloured triangles along the vertical axis on the lower plots in figure 12 indicate the foil modal frequencies (also shown in figure 3), which are higher in the FV flow than in the FW flow because of a reduction in added mass when the majority of the suction side is enclosed in the ventilated cavity. The results show transition from FW to FV flow has an impulsive, broad-banded excitation in the $\theta$ time-frequency spectra early in the acceleration process for the CW run 4801. The dominant frequency observed for the FW wave run 5001 is the wave encounter frequency ($\,f_e$). The time-frequency spectra of both the CW and wave runs also show peaks at the foil modal frequencies ($\,f_{1-4}$), the carriage modal frequency ($\,f_{car}$) and the vortex shedding frequency ($\,f_{vs}$). The vortex shedding frequency ($\,f_{vs}$) increases and decreases in the acceleration and deceleration regions, respectively, following from the constant Strouhal scaling of (4.2).

For the FV CW case on the left, a relatively sharp peak can be observed in the power spectra plot in figure 12 at $\,f_{vs}$ because of lock-in with the nearest foil subharmonic frequency, $\,f_3/2$, which increases the intensity of the twist fluctuations at $\,f_{vs}$ and at $\,f_3$. The influence of subharmonic lock-in can also be observed in the underwater photographs, where the distance between the aerated von Kármán vortex street aft of the foil trailing edge for the FV case appears to be half that of the FW case on the right in figure 10. However, the vortex spacing should be the same since $\,f_{vs}$ is the same for both cases with the same $F_{nh}$, $\alpha$ and $AR_h$. One possible explanation for the differences in vortex spacing is lock-in of $\,f_{vs}$ with $\,f_3/2$, and excitation of $\,f_3$ leads to shedding of additional vortices into the wake at twice the nominal vortex shedding frequency – effectively halving the spacing.

6.1.3. Influence of $\alpha$ and waves: $AR_h = 1, F_{nh} = 1.5$

Figure 13 shows the effect of $\alpha$ on hysteresis loops for $AR_h = 1$, $F_{nh} = 1.5$ for CW cases on the left and wave cases on the right. To avoid overcrowding the graph, results are limited to representative cases that include both FW and FV conditions. The hysteretic effects of ventilation can be visualized by plotting the moving time averages of $C_L$ and $C_M$ against the instantaneous Froude number $F_{nh}$. If the flow is FW throughout the run, such as for $\alpha \le 14^\circ$ in both CW and wave cases, the acceleration and deceleration phases are nearly identical, i.e. no hysteresis is apparent. Comparison of the left and right columns of figure 13 shows that if the flow remains FW, the hysteresis loops are also unaffected by the presence of monochromatic non-breaking waves, i.e. the moving time-average values are the same for CW and wave cases with the same $\alpha$, $F_{nh}$ and $AR_h$. For $\alpha = 15^\circ$, the foil is FV in CW but FW in waves, as observed in the underwater snapshots in figure 10. Hence, the hysteresis loops in figure 13 show the reduction in $C_M$ for the FV CW case on the left (as indicated by the thick purple line) relative to the FW wave case on the right (as indicated by the thin purple line). However, the difference in $C_L$ between the left and right plots is negligible. As noted earlier, $C_M$ is more sensitive to ventilation than $C_L$ due to a change in the centre of pressure. In addition, as observed in figures 6 and 13, the change in load coefficients and resultant hysteresis effect are small for $AR_h = 1$ and $F_{nh} = 1.5$.

Figure 13. Influence of $\alpha$ on the hysteresis loops formed by the moving time-averaged response of the lift ($C_L$) and moment ($C_M$) coefficients versus the instantaneous Froude number $F_{ni}$, $F_{nh}=1.5$, $AR_h=1.0$, $P_t=P_{atm}$. The left and right plots show the CW and wave ($T_w=1.5$ s, $A_w=0.05$ m) runs, respectively. The FW and FV results are shown as thin and thick solid lines, respectively, and are indicated in the line legend. The mean FW loads increase with higher $\alpha$ and plateau when ventilation develops, such as observed for $C_M$ at $\alpha = 15^\circ$ for CW on the left.

The PSDs of the fluctuating $\delta$ and $C_L$ for runs with varying $\alpha$ in CW and in waves for $F_{nh}=1.5$, $AR_h=1.0$, $P_t=P_{atm}$ are shown in figure 14. A comparison of the top (CW) and bottom (waves) plots of the PSD of the fluctuating $C_L$ (left) and $\delta$ (right) shows that waves add a peak at the wave encountered frequency $\,f_{e}$, but have minimal impact on the rest of the frequency response. Waves again produce a ‘spreading’ effect upon the third resonance ($\,f_3$), caused by the modulation in the natural frequency with the periodic change in immersion depth. In this, the difference between FV and FW flows is small, due to the relatively small size of the ventilated cavity at $F_{nh}=1.5$.

Figure 14. The effect of $\alpha$ on the PSD of the fluctuating lift coefficient ($C_L$) on the left and normalized tip bending displacement ($\delta /c$) on the right for run conditions $F_{nh}=1.5$, $AR_h=1.0$, $P_t=P_{atm}$. The top and bottom plots show the CW and wave ($T_w=1.5$ s and $A_w=0.05$ m) runs, respectively. The FW and FV results are shown as thin and thick solid lines, respectively, as indicated in the line legend. The carriage ($\,f_{car}$) and FW foil modal frequencies ($\,f_{1-4,FW}$) are indicated by the vertical green dotted and red dashed lines, respectively. The FV foil modal frequencies are slightly higher but are not shown to avoid over-cluttering the graph. The wave encounter frequency is $\,f_e=1.37$ Hz. The results show that the PSDs are not sensitive to variations in $\alpha$, but small differences can be observed between the FW and FV runs.

6.2.1. Calm water: $\alpha = 5^\circ, \alpha = 13^\circ$, & $\alpha = 20^\circ$

Figure 15 shows the above water and underwater views of three runs with different angles of attack ($\alpha = 5^\circ, \alpha = 13^\circ, \alpha = 20^\circ$) for atmospheric conditions at $AR_h = 2$ and $F_{nh} = 1.5$ in CW. Figures 16 and 17 depict the time histories and PSDs of the load and deformation fluctuations, respectively, for the same three runs.

Figure 15. Above water view (top) of the surface pattern and underwater view (bottom) of the suction side of the foil for $\alpha =5^\circ$ (a), $13^\circ$ (b) and $20^\circ$ (c) in CW condition. Here $AR_h = 2$, $F_n = 1.5$, $P = P_{atm}$. The flow is FW for $\alpha = 5^\circ$ and $13^\circ$, and FV for $\alpha = 20^\circ$. (a) CW, Run 2202, $\alpha =5^\circ$. (b) CW, Run 8101, $\alpha =13^\circ$. (c) CW, Run 3601, $\alpha =20^\circ$.

Figure 16. Time histories of the raw hydrodynamic load coefficients ($C_L, C_D, C_M$) and tip deformations ($\delta /c, \theta$) for $\alpha =5^\circ$ (FW), 13$^\circ$ (FW) and 20$^\circ$ (FV) in CW. Here $AR_h = 2$, $F_n = 1.5$, $P = P_{atm}$. The horizontal black dashed lines indicate the average of the values in the steady-speed region. Compared with the results for $\alpha =5^\circ$ shown in figure 8(a), the mean load coefficients in the left are slightly higher because of a reduction in 3-D losses with the higher $AR_h$, but the amplitude of load fluctuations are much higher because of dynamic load amplification caused by frequency coalescence of modes 2 and 3, as suggested in figure 3. The amplitude of the fluctuations is lower for $\alpha = 13^\circ$ compared with $\alpha = 5^\circ$, because energy is pulled away from the coalescence frequency to the nearby vortex shedding frequency, as confirmed later in figure 18. The $\alpha = 20^\circ$ case shows ventilation-induced stabilization, which resulted in the smallest amplitude of fluctuations among the three cases. (a) CW, Run 2202, $\alpha =5^\circ$. (b) CW, Run 8101, $\alpha =13^\circ$. (c) CW, Run 3601, $\alpha =20^\circ$.

Figure 17. Power spectral densities (PSD) of the hydrodynamic load coefficients ($C_L, C_D, C_M$) and tip deformations ($\delta /c, \theta$) for a $\alpha =5^\circ$, $13^\circ$ and $20^\circ$ in CW. Here $AR_h = 2$, $F_n = 1.5$, $P = P_{atm}$. The carriage $\,f_{car}$ and foil modal frequencies $\,f_{1-4}$ are indicated by the vertical green and blue dashed-dotted lines, respectively. The wave encounter frequency $\,f_{e}$ and the vortex shedding frequency $\,f_{vs}$ are indicated by the vertical red and orange dashed lines, respectively. For $\alpha = 5^\circ$, a significant peak appears at 28 Hz due to coalescence of $\,f_2$ and $\,f_3$. The peak at 28 Hz is substantially reduced for $\alpha = 13^\circ$ due to energy being pulled to nearly $\,f_{vs}$. The amplitude near 28 Hz is further reduced at $\alpha = 20^\circ$ due to the transition to FV flow, which caused $\,f_2$ and $\,f_3$ to be higher and to the left and right of $\,f_{vs}$, respectively. (a) CW, Run 2202, $\alpha =5^\circ$. (b) CW, Run 8101, $\alpha =13^\circ$. (c) CW, Run 3601, $\alpha =20^\circ$.

As observed in figure 15, the flow is FW for $\alpha = 5^\circ$ and $13^\circ$, and FV for $\alpha = 20^\circ$. Comparison between figure 8(a) for $AR_h=1$ and figure 16(a) for $AR_h=2$, both at $\alpha = 5^\circ$ in CW, confirm that the mean load coefficients are slightly higher for $AR_h=2$ than for $AR_h=1$ because of reduced 3-D losses. However, the amplitudes of fluctuations in the loads and deformations are much higher for $AR_h=2$ than for $AR_h=1$, due to frequency coalescence of modes 2 and 3, which causes co-resonance of the two modes and enhanced dynamic amplification of nearby excitations. As observed in the PSDs shown in figure 17(a) and the time-frequency spectra shown in figure 18(a), the largest peak for $\alpha = 5^\circ$ occurs at about 28 Hz, where modes 2 and 3 both reside. The time-frequency spectrum in figure 18(a) shows energy concentrated along a line of constant Strouhal number, representing $\,f_{vs}$ in the acceleration and deceleration region. At the steady-speed region, this peak overlaps the coalescent modes 2 and 3 at 28 Hz, although a smaller peak can also be observed at $\,f_{vs}$.

Figure 18. Time-frequency spectra of the tip twist deformation for $\alpha =5^\circ$, $13^\circ$ and $20^\circ$ in CW. Here $AR_h = 2$, $F_n = 1.5$, $P = P_{atm}$. Each subplot has three panels. The top panel shows the time history of $\theta$ (solid blue line) and the instantaneous Froude number ($F_{ni}$) (red dashed line). The lower left panel indicates the power spectra (PS) and the lower right panel indicates the time-frequency spectra of $\theta$. The cyan-coloured triangles on the $y$ axis of the lower two panels mark the first three modal frequencies of the hydrofoil. The modal frequencies are higher in the FV flow ($\alpha = 20^\circ$ case) than in the FW flow ($\alpha = 5^\circ$ and $13^\circ$ cases). The carriage frequency ($\,f_{car}$) is indicated by the green crosses. The vortex shedding frequency ($\,f_{vs}$) is indicated by the magenta dashed line in the lower panels. Intense energy concentration can be observed at 28 Hz due to coalescence of $\,f_2$ and $\,f_3$ for $\alpha = 5^\circ$. Energy modulates between $\,f_2, f_3$ and $\,f_{vs}$ for $\alpha = 13^\circ$, which is responsible for the lower amplitude of fluctuations compared with $\alpha = 5^\circ$. For $\alpha = 20^\circ$, energy concentration jumps to 50 Hz in the FV flow. (a) CW, Run 2202, $\alpha =5^\circ$. (b) CW, Run 8101, $\alpha =13^\circ$. (c) CW, Run 3601, $\alpha =20^\circ$.

At $\alpha = 13^\circ$, the proximity of $\,f_{vs}$ to $\,f_2, f_3$ results in frequency and amplitude modulations between these frequencies, as evident in figures 17(b) and 18(b), which lead to lock-off and reduced amplitude of load and deformation fluctuations compared with $\alpha = 5^\circ$. Contributing to this is the enhancement of viscous effects within the leading edge vortex, its associated shear layer and the wake of the hydrofoil at higher attack angles. Viscous damping is thought to increase with the size and strength of these shear layers, reducing the energy exchange between proximate modes and the vortex shedding.

With the stable FV flow at $\alpha = 20^\circ$ comes significantly reduced amplitude of fluctuations in hydrodynamic loads and deformations compared with FW flows at smaller attack angles, as observed in figure 16. This occurs because (1) $\,f_2$ and $\,f_3$ are more separated in the FV flow, as noted earlier in this subsection and shown in figure 3; and (2) the FV flow tends to lead to broad-band excitation between 25–55 Hz, reducing the concentration of energy at $\,f_{vs}$ as the vortex street is obliterated over most of the submerged span. Following the transition to the FV flow (around the 10-s mark), the dominant peak in the time-frequency shifts from the $\,f_{vs}$ to near 50 Hz, as shown in figure 18(c). An energy concentration at $\,f_{vs}$ is observed again in the deceleration stage, as the flow reattaches and transitions back to the FW flow.

The SSI-SPOD co-analysis provides additional insight into the concurrent structural and fluid dynamic modes. The results of the co-analysis for the same three runs (trials 1-2202, 1-8101 and 1-3601) are tabulated in table 3.

Table 3. Tabulations of poles identified by SSI. Shown are the undamped natural frequency ($\,f_0$), damped natural frequency ($\,f_n$) and total (structural $+$ hydrodynamic) damping ratio ($\xi$, as a percentage of critical damping) for $\alpha =5^\circ$ (left), $13^\circ$ (middle) and $20^\circ$ (right) in CW. Here $AR_h = 2$, $F_n = 1.5$, $P = P_{atm}$. A combination of the structural mode shapes and concurrent hydrodynamic modes (shown in figures 19–21) were used to classify each pole as an excitation (E) or resonance (R). Spurious modes and harmonics are indicated by shaded rows. The undamped frequencies are physically meaningful only in the cases of structural resonances so they are omitted for spurious modes or forced vibration. Note the very low total damping ratio of 0.06 % when modes 2 and 3 coalesce for run 1-2202 with $\alpha = 5^\circ$.

The results of the co-analysis, which includes fitted structural PSDs with identified modes and concurrent hydrodynamic and structural modes, are presented in the following figures: run 1-2202 in figure 19, run 1-8101 in figure 20 and run 1-3601 in figure 21.

Figure 19. Results for run 1-2202; $\alpha =5^\circ$, $F_{nh}=1.5$, $AR_h=2.0$, FW flow, CW. (a) Tip bending and twisting auto-power spectral densities (measured and fitted with SSI). Blue dash-dot lines indicate the modal frequencies anticipated from historical results. The red dashed line is the theoretical $\,f_{vs}$ ($St=0.265$). Black dashed lines indicate modes fitted using the SSI algorithm. (b) Results of co-analysis. The SPOD modes (top) show visually observable hydrodynamic (SPOD) modes and SSI modes (bottom) show the structural modes of the fitted model at the same frequencies. Structural modes are not generally accompanied by coherent hydrodynamic modes, while forced-vibration responses like that at the vortex shedding frequency show clearly defined hydrodynamic modes. The dominant peak is near 28.6 Hz, where modes 2 and 3 coalesce. A secondary peak can also be observed at its harmonic near 57 Hz. Movie 1 in the supplementary material contains animations of the complex modes.

Figure 20. Results for run 1-8101: $\alpha =13^\circ$, $F_{nh}=1.5$, $AR_h=2.0$, FW flow, CW. (a) Tip bending and twisting auto-power spectral densities (measured and fitted with SSI). (b) Results of co-analysis, showing snapshots of hydrodynamic (top) and structural (bottom) modes. At a larger attack angle, the coherence of the wake structure breaks down, eliciting a much less energetic response in the coalescent modes 2 and 3, as visible by comparing with figure 19 at $\alpha = 5^\circ$. Movie 2 in the supplementary material contains animations of the complex modes.

Figure 21. Results for run 1-3601: $\alpha =20^\circ$, $F_{nh}=1.5$, $AR_h=2.0$, FV flow, CW. (a) Tip bending and twisting auto-power spectral densities (measured and fitted with SSI). (b) Results of co-analysis, showing snapshots of hydrodynamic (top) and structural (bottom) modes. With ventilation present, the natural frequencies of all modes is expected to increase as the added mass is reduced. The apparent locations of the second and third structural resonances are much higher than anticipation, which suggests a deficiency in the empirical model used to predict FV modal added-mass coefficients. Movie 3 in the supplementary material contains animations of the complex modes.

The results for run 1-2202 ($\alpha =5^\circ$, $F_{nh}=1.5$, FW flow, CW) in figure 19 illustrate the dynamical behaviour of the hydrofoil at a low angle of attack. The top plot (a) shows the measured auto-power spectral densities for $\delta$ and $\theta$. Bold overlaid lines represent the spectra of the fitted SSI model. Vertical, blue, dashed-dotted lines indicate the resonant frequencies anticipated from prior experiments and empirical models (shown in figure 3). A red dashed line indicates the predicted $\,f_{vs}$. Solid, black, vertical lines indicate the poles fitted by the SSI algorithm – each of which may represent a resonant mode, an energetic response to a periodic hydrodynamic excitation or an artifact of over-fitting. In figure 19(b) the upper row depicts snapshots of the complex SPOD modes at frequencies matching the vertical black lines. The lower row depicts the experimental structural modes (SSI modes) at those same frequencies as contours of the hydrofoil's displacement normal to its planform, with bolded contours indicating node lines. Each column of figure 19(b) can thus be interpreted as a snapshot of the synchronous hydro-structural mode at the indicated frequency. A combination of the mode shapes and the associated flow patterns were used to classify each identified pole as a structural resonance, the forced response to external excitation or a spurious artifact. It should be noted that poles with damping ratios exceeding 50 % of critical damping are omitted to conserve space (as they are reasoned to be unphysical); as a result, certain poles indicated by vertical black lines in figure 19(a) may not be depicted.

In general, the SSI poles correspond closely to the predicted frequencies of the first three resonances and vortex shedding. In figure 19 the first pole shows a first bending mode, with some fluctuations in the foil's aerated base cavity as a result. The FW flow and small attack angle produce coalescence between modes 2 (first twisting) and 3 (second bending) at about 28.6 Hz, where the structural mode shows combined bending and torsion and the total critical damping ratio drops to 0.06 % (table 3). At the same frequency, there is some evidence of vortex shedding in the wake of the hydrofoil, though breaks in the vertical striations near the free surface (figure 19b) suggest that a component of the visual mode is attributable to rippling in the base cavity produced by structural vibration. In both cases, the SPOD modes do not show enough coherence to suggest that the peak in the structural spectrum of figure 19(a) originates with some hydrodynamic mode; rather, the co-analysis results suggest that structural resonances are the cause and that the SPOD modes illustrate the effects of the structural motion upon the visual flow field. The final pole is attributed to vortex shedding, with coherent striations along the foil TE and parallel to the closure line of the base cavity, which indicate bubbly entrainment into alternating vortex cores. The associated pressure fluctuations produce an operational deflection shape (ODS) dominated by torsion, which is classified as (non-resonant) forced vibration produced by the hydrodynamic excitation. Energy is shared with the coalescent modes by the nearby vortex shedding frequency, although the existence of two distinct peaks suggests that lock-in has not occurred. The pole at 25 Hz is judged to be spurious because it possesses low energy in the preceding spectra, it does not correspond to any known resonances, and it presents no coherent flow pattern.

As the attack angle is increased in figure 20, the most significant change is the reduction in peak energy at modes 2 and 3. Table 3 shows a huge increase in the damping of the combined modes; some of this is likely physical and owed to the increase in viscous effects in the wake and shear layers mentioned earlier. At the same time, it may signify that excitation at proximate frequencies has been suppressed, evident in the suppression of the vortex shedding peak and the lack of coherent striation in the SPOD mode. The structural ODS associated with vortex shedding has a more pronounced torsion component than at the lower attack angle, which may be due to the increased angle between the foil section and the flow-aligned wake. Additionally, the frequency of the first bending mode increases slightly, moving further above the anticipated first modal frequency because, as seen in figure 15, the higher angle of attack produces a deeper depression along the (visible) suction surface of the hydrofoil, reducing slightly the effective immersion depth and associated added mass.

Finally, for the FV flow shown in figure 21, the character of the spectra are altered substantially. Simulation and physics-based modelling (Harwood et al. Reference Harwood, Felli, Falchi, Garg, Ceccio and Young2020; Young et al. Reference Young, Wright, Yoon and Harwood2020) suggest that modes 2 and 3 should separate and reverse their order as a result of their disparate changes in hydrodynamic added mass in the FV flow, but no peaks appear at the predicted frequencies. Those FV modal frequency predictions have not previously been validated against experiments at such large immersed aspect ratios however, so their veracity cannot be confirmed. The two closely spaced peaks at about 34 and 36 Hz appear to show a mixture of bending and twisting in the structural modes, with the former showing greater bending and the latter more twisting. The lack of evident vortex shedding in the SPOD modes or known sources of excitation leads us to conclude that these are, respectively, mode 3 (second bending) and mode 2 (first twisting). The differences between mode shapes, while subtle, suggest that the mode order has reversed as expected, and the disparity between these frequencies and those predicted can be attributed to the inaccuracy of the modal added-mass model used to make the predictions. Another notable result is a similarity of the first resonant frequency to that in the FW flow. As shown in table 3, the resonant frequency of mode 1 actually decreases slightly with the transition to the FV flow, contrary to the findings of Harwood et al. (Reference Harwood, Felli, Falchi, Garg, Ceccio and Young2020). The exchange of liquid water for air should reduce the modal added mass, but the large angle of attack causes a significant deformation in the free surface. This includes a large run-up on the pressure side of the hydrofoil, which would increase the fluid added mass. Moreover, the expected change in resonant frequency is only approximately 0.5 Hz – small enough that it likely falls within the fitting error of the output-only SSI algorithm.

6.2.2. Waves: $\alpha = 5^\circ$, $\alpha = 13^\circ$, & $\alpha = 20^\circ$

The time-frequency spectra of the tip twist deformations ($\theta$) for $\alpha = 5^\circ$, $\alpha = 13^\circ$, and $\alpha = 20^\circ$ in waves are shown in figure 22. Just as in CW, the flow is FW for $\alpha = 5^\circ$ and $13^\circ$, and FV for $\alpha = 20^\circ$. Since the flow conditions are the same as those in the preceding section, snapshots from the above and underwater videos are omitted.

Figure 22. Time-frequency spectra of the tip twist deformation ($\theta$) for $\alpha = 5^\circ, 13^\circ$ and $20^\circ$ in waves. Here $AR_h = 2$, $F_n = 1.5$, $P = P_{atm}$. Each subplot has three panels. The top panel shows the time history of $\theta$ (blue solid line) and the instantaneous Froude number ($F_{ni}$) (red dashed line). The lower left panel indicates the power spectra (PS) and the lower right panel indicates the time-frequency spectra of $\theta$. The cyan-coloured triangles on the $y$ axis of the lower two panels mark the first three modal frequencies of the hydrofoil ($\,f_{1-3}$). The carriage frequency ($\,f_{car}$) is indicated by the green crosses. The vortex shedding frequency ($\,f_{vs}$) is indicated by the magenta dashed line and the wave encounter frequency ($\,f_{ex}$) is indicated by the red dotted line in the lower panels. Comparisons with the corresponding CW runs in figure 18 show a similar dynamic response except for the addition of a significant peak at $\,f_e$ and wave-induced modulation of the vortex shedding frequency and modal frequencies. (a) CW, Run 2402, $\alpha =5^\circ$. (b) CW, Run 8301, $\alpha =13^\circ$. (c) CW, Run 3801, $\alpha =20^\circ$.

The slowly moving mean is the same between the CW and wave cases with the same $\alpha$, as observed by comparing the mean $\theta$ in the steady-speed region in the top panels in figures 18 and 22. Waves produce modulations of the foil modal frequencies and vortex shedding frequency visible as small peaks at $\,f_2 \pm f_e$, $\,f_3 \pm f_e$ and $\,f_{vs} \pm f_e$ in the power spectra and time-frequency spectra shown at the bottom of figure 22 for all three wave cases, but not the CW cases shown in figure 18. Waves did not reduce the large fluctuations in the hydrodynamic loads and deformations when modes 2 and 3 coalesced at $\alpha = 5^\circ$, as observed in figure 22(a).

6.2.3. Influence of $\alpha$ and waves: $AR_h = 2, F_{nh} = 1.5, P_t = P_{atm}$

The hysteresis loops formed by the moving time-averaged response of the lift and moment coefficients ($C_L, C_M$) versus the instantaneous Froude number ($F_{ni}$) for $AR_h = 2$, $F_{nh} = 1.5$, $P_t = P_{atm}$ for cases in CW and waves are shown on the left and right plots, respectively, in figure 23. The PSDs of the fluctuating $C_L$ are shown on the left and $\delta /c$ on the right, for CW on the top and waves on the bottom, are shown in figure 24. The flow conditions (FW or FV) and $\alpha$ are indicated in the line legends. To avoid over-cluttering the graphs, only representative cases in the FW and FV regimes are shown in the two figures to illustrate the influence of $\alpha$ and waves.

Figure 23. Hysteresis loops formed by the moving time-averaged response of the lift and moment coefficients ($C_L, C_M$) versus the instantaneous Froude number ($F_{ni}$) for varied $\alpha$. Here $AR_h = 2, F_{nh} = 1.5, P_t = P_{atm}$. The (a,c) and plots show the CW and wave ($T_w=1.5$ s, $A_w=0.05$ m) runs, respectively. The FW and FV results are shown as thin and thick solid lines, respectively. The moving average results are very similar between the CW and wave runs with the same $\alpha$. A significant drop in $C_m$ is observed in the FV flow at $\alpha = 20^\circ$.

Figure 24. Power spectral density of the fluctuating lift coefficient ($C_L$) on the left and tip bending displacement ($\delta /c$) on the right. Results shown for varied $\alpha$. Here $AR_h = 2$, $F_{nh} = 1.5$, $P_t = P_{atm}$. The top and bottom plots show the CW and wave ($T_w=1.5$ s, $A_w=0.05$ m) runs, respectively. The FW and FV results are shown as thin and thick solid lines, respectively. The carriage ($\,f_{car}$) and the FW foil modal frequencies ($\,f_{i,FW},\ i=1,2,3,4$) are indicated by the vertical green and red dashed lines, respectively. The FV foil modal frequencies ($\,f_{i,FV},\ i = 1,2,3,4$) are slightly higher, but are omitted to avoid over-cluttering the graph. The vortex shedding frequency ($\,f_{vs}$) is shown as the vertical maroon dashed line. The wave encounter frequency ($\,f_{e}$) is 1.67 Hz. For CW and wave conditions at $\alpha = 5^\circ$, the most significant peak is at 28 Hz caused by coalescence of modes 2 and 3 of the foil and a lower peak is also visible at its harmonic at 56 Hz. The proximity of the vortex shedding frequency ($\,f_{vs}$) to $\,f_{2}$ and $\,f_{3}$ lead to frequency demodulation and de-coalescence, which reduce the amplitude at 28 Hz and broaden the peak at higher $\alpha$'s. At $\alpha = 20^\circ$ for the FV flow, ventilation-induced stabilization is observed. In waves the most significant peak is at $\,f_e$ for all the cases.

As shown in figure 23, the hysteresis loop of the slowly moving mean response is unchanged by the presence of waves. The moment coefficient of the FV run with $\alpha = 20^\circ$ is lower, while the lift coefficient is slightly higher than the FW runs with $\alpha = 12^\circ, 13^\circ$ in both CW and waves. For the FV runs, small differences can be observed between the hysteresis response in the CW case on the left and the wave case on the right in figure 23. The small difference can be attributed to a delay in flow reattachment in the wave case.

As shown in figure 24, at $\alpha = 5^\circ$ for the FW flow, the majority of the energy is concentrated around 28 Hz, where modes 2 and 3 have nearly coalesced. It appears that as $\alpha$ increases, the energy of the vortex shedding is reduced, and with it, the energy of the nearby coalescent modes is reduced as well. Compared with figure 14 shown in § 6.1 for $AR_h=1$, the peak at mode 1 is not as prominent in $AR_h=2$ compared with $AR_h=1$ because hydrodynamic damping tends to increase with submergence and speed.

The results shown in figure 14 for $AR_h = 1$, $F_{nh} = 1.5$ and figure 21 for $AR_h = 2$, $F_{nh} = 1.5$ both exhibit energy peaks between 40–50 Hz. Energy peaks between 40–50 Hz can also be observed for test runs with different speeds ($F_{nh}$) with $AR_h = 1$ in figure 22. Given that the energy peaks between 40–50 Hz are independent of $\alpha$, $AR_h$ and $F_{nh}$, the source of the peaks is most likely due to the foil lead-lag mode ($\,f_4 = 68.4$ Hz, also called surge mode) exciting the higher order surge mode frequencies in the carriage between 40–50 Hz. The results in figures 14 and 21 show that the intensity of peaks between 40–50 Hz increase in the FV flow. This is because roll-up of the aerated cavity on the suction side impacts the foil TE (which can be observed in figures 15c, 21 and 27), and hence, increases the intensity of the lead-lag excitation and the higher order surge modes of the carriage.

Once again, we can further explore these hypothesized effects through the lens of the SSI-SPOD co-analysis. A summary of the modal analysis is shown in table 4. Shown in figures 25, 26 and 27 are the output PSDs and co-analysis modes for the runs at $\alpha =5^\circ$, $\alpha =13^\circ$ and $\alpha =20^\circ$, respectively, in waves. In all three figures, the principal difference when compared with figures 19–21 is the presence of a well-defined peak at the wave encounter frequency. Additionally, secondary modes appear, flanking the mean resonant frequencies of certain modes as a result of the wave modulation. The intensities of the secondary side lobes are typically small, so the ability of the SSI algorithm to fit them is tenuous.

Figure 25. Results for run 1-2402: $\alpha =5^\circ$, $F_{nh}=1.5$, $AR_h=2.0$, FW flow, waves. (a) Tip bending and twisting auto-power spectral densities (measured and fitted with SSI). (b) Results of co-analysis, showing snapshots of hydrodynamic (top) and structural (bottom) modes. Modes 2 and 3, while coalescent near 28 Hz, show significant modulation in their frequencies by the wave encounter frequency ($\pm f_e$), as well as interaction with the foil TE vortex shedding frequency, $\,f_{vs}$. Movie 4 in the supplementary material contains animations of the complex modes.

Figure 26. Results for run 1-8301: $\alpha =13^\circ$, $F_{nh}=1.5$, $AR_h=2.0$, FW flow, waves. (a) Tip bending and twisting auto-power spectral densities (measured and fitted with SSI). (b) Results of co-analysis, showing snapshots of hydrodynamic (top) and structural (bottom) modes. Vortex structures in the TE wake are far less coherent than at smaller attack angles (e.g. $\alpha = 5^\circ$ in figure 25). Movie 5 in the supplementary material contains animations of the complex modes.

Figure 27. Results for run 1-3801: $\alpha =20^\circ$, $F_{nh}h=1.5$, $AR_h=2.0$, FV flow, waves. (a) Tip bending and twisting auto-power spectral densities (measured and fitted with SSI). (b) Results of co-analysis, showing snapshots of hydrodynamic (top) and structural (bottom) modes. Wave modulation and reduced intensity due to de-coalescence and ventilation-induced stabilization suppressed the distinct peaks for mode 2 and mode 3, causing only a single mode to be fitted. Movie 6 in the supplementary material contains animations of the complex modes.

Table 4. Tabulations of the undamped natural frequency ($\,f_0$), damped natural frequency ($\,f_n$) and percent total (structural $+$ fluid) critical damping ($\xi$) for each of the modes produced by SSI for $\alpha = 5^\circ, 13^\circ$ and $20^\circ$ in waves. Here $AR_h = 1$, $F_n = 1.5$, $P = P_{atm}$. A combination of the structural mode shapes and concurrent hydrodynamic modes (shown in figures 19–27) were used to classify each mode as an excitation (E), resonance (R) or a spurious mode. Compared with the values for CW reported in table 3, the damping coefficients seem slightly higher in general. In particular, at $\alpha = 5^\circ$ for run 2402, $\xi = 0.58\,\%$ when modes 2 and 3 coalesced, compared with 0.06 % for the CW case in table 3. The exception is the wave encounter frequency, where a negative damping is fitted for run 1-2402. While the quantitative value of the damping estimates are limited by the output-only analysis, the small positive or negative damping ratios indicate excess energy in the input spectrum, concentrated at a single frequency.

In figure 25 the wave encounter produces an ODS with low-order bending. Mode 1 is unaffected by the presence of waves, but additional peaks appear between 25 and 35 Hz, where the peak at the coalescence frequency of 28.8 Hz is accompanied by side lobes at approximately 27.1 and 30.4 Hz – both separated from the mean peak by a distance equal to the wave encounter frequency $\,f_e$. Additionally, the SPOD modes show significant wake vortex structures at 28.8 and 30.39 Hz, which suggests that one or both of modes 2 and 3 experienced intermittent lock-on with the vortex shedding mode as a result of the wave modulation band and the proximity to the $\,f_{vs}$. The vortex shedding mode itself, while still showing signs of a Von Kàrmàn street in the wake, appears less coherent than at the slightly lower frequencies. As $\alpha$ increases in figure 26, the wake structures break down, suggesting again that the TE vortex shedding is less operative at larger attack angles. While no modulation is explicitly detected at $\alpha =13^\circ$, this is a result of the reduced spectral power in the central peak, with no distinct side lobes evident. Instead, the increased damping of the mode fitted to the central peak speaks to the modulation of that peak's location. Accompanying the reduced vortex shedding power, the coalescent modes 2 and 3 show a much reduced peak power, as was the case in CW. Moreover, the apparent interactions between mode 2, mode 3 and the vortex shedding mode at $\alpha =5^\circ$ are gone. As the angle of attack is further increased to $\alpha = 20^\circ$ and the FV flow is realized in figure 27, only a single peak is detected at about 35 Hz, whereas two were detected in figure 21 for the CW case. The low peak power, close spacing of the two modes and the wave modulation combine to result in only a single mode being fitted with an erroneously large damping ratio. Interestingly, a single side lobe is detected for mode 1.