4.1. Modal frequencies

To verify the system parameters used in the ROMs presented in § 3, the predicted system modal frequencies are compared with experimental measurements of the SS and CF hydrofoils in fully wetted conditions in figure 5. The experimental measurements of $C_N^{\prime }$ were averaged for incidences ($\alpha _o$) ranging from $0^{\circ }$ to $14^{\circ }$ in increments of $2^{\circ }$ in non-cavitating (fully wetted or FW) conditions at $Re=0.6\times 10^{6}$. The average power spectral density (PSD) was used, as the frequency at the peak of the PSD varied slightly with $\alpha _o$ owing to changes in entrained fluid inertia, particularly when stall develops for $\alpha _o \gtrsim 10^{\circ }$ (Zarruk et al. Reference Zarruk, Brandner, Pearce and Phillips2014; Young et al. Reference Young, Garg, Brandner, Pearce, Butler, Clarke and Phillips2018). No frequency modulations were observed in fully wetted conditions. It should be noted that in the experimental studies presented by Zarruk et al. (Reference Zarruk, Brandner, Pearce and Phillips2014), the same hydrofoils with the same mounting set-up and force balance (FB) were used in the same cavitation tunnel. The FB frequency was 122 Hz, which is significantly higher than the first FW bending frequency, $\,f_{1,FW}$, of 58 Hz for the SS hydrofoil and 41 Hz for the CF hydrofoil. As noted in § 3, the dynamic FSI response was modelled using a 1-DOF forced parametric oscillator model considering spanwise bending only. The dynamic model can be extended to 2-DOF, which would yield dry ($\,f_{n2,dry}$) and fully wetted ($\,f_{n2,FW}$) second mode (twisting) frequencies as presented in table 3. The $\,f_{n2,FW}$ is also shown in figure 5.

Figure 5. Measured (Exp) averaged power spectral density (PSD) of the fluctuating normal force coefficients ($C_N^{\prime }$) for the SS and CF hydrofoils for incidences ($\alpha _o$) ranging from $0^{\circ }$ to $14^{\circ }$ in increments of $2^{\circ }$ in non-cavitating conditions at $Re=0.6\times 10^{6}$ (Zarruk et al. Reference Zarruk, Brandner, Pearce and Phillips2014). The vertical lines correspond to the predicted first and second system bending natural frequencies in fully wetted condition, $\,f_{1,FW}$ and $\,f_{2,FW}$, for the SS and CF hydrofoils, their visible harmonics and the measured averaged force balance (FB) natural frequency, $\,f_{FB}$. Good agreement is observed between the predicted modal frequencies and the location of the peaks of the frequency spectra, especially considering that the measured PSD varies slightly with $\alpha _o$ owing to changes in entrained fluid inertia, particularly when stall develops with $\alpha \gtrsim 10^{\circ }$ (Zarruk et al. Reference Zarruk, Brandner, Pearce and Phillips2014; Young et al. Reference Young, Garg, Brandner, Pearce, Butler, Clarke and Phillips2018).

As shown in figure 5, good agreement was observed between the predicted modal frequencies and the location of the peaks of the averaged PSD of the $C_N^{\prime }$ reported by Zarruk et al. (Reference Zarruk, Brandner, Pearce and Phillips2014). Figure 5 demonstrates that the FW twisting frequency was much higher than the FW bending frequency, and hence the separation of time scale justifies the use of the 1-DOF dynamic model considering the bending degree of freedom only.

4.2. Cavitation patterns

Examples of the observed cavitation patterns of the SS and CF hydrofoils at selected cavitation numbers are shown in figures 6 and 7, respectively. The corresponding variation of the measured maximum local attached cavity length normalized by the local chord at each section ($L_{c}(y)/c(y$)) with $\sigma$ for the SS and CF hydrofoils are shown in open circles and cyan filled squares, respectively, in figure 8. The predicted cavity length normalized by the mean chord ($L_{c}/c$) obtained using (3.8) for the SS and CF hydrofoils are shown as a blue dashed line and a magenta dash–dotted line, respectively.

Figure 6. Typical example images of cavitation of the SS hydrofoil at selected values of $\sigma$. (a) Type I shock-wave-driven cavity shedding only at $\sigma = 0.3$.(b–d) Combined Type II re-entrant jet and Type I shock-wave-driven cavity shedding for $0.4 \lesssim \sigma \lesssim 0.6$. (e–h) Type II re-entrant jet cavity shedding only for $0.7 \lesssim \sigma \lesssim 1.0$. Tip vortex cavities can be observed for $0.3 \lesssim \sigma \lesssim 0.7$ in panels (a–e): (a) $\sigma =0.3$; (b) $\sigma =0.4$; (c) $\sigma =0.5$; (d) $\sigma =0.6$; (e) $\sigma =0.7$; ( f) $\sigma =0.8$; (g) $\sigma =0.9$; (h) $\sigma =1.0$.

Figure 7. Typical example images of cavitation of the CF hydrofoil at selected values of $\sigma$. (a) Type I shock-wave-driven cavity shedding only at $\sigma = 0.3$. (b–d) Combined Type II re-entrant jet and Type I shock-wave-driven cavity shedding for $0.4 \lesssim \sigma \lesssim 0.6$. (e–h) Type II re-entrant jet cavity shedding only for $0.7 \lesssim \sigma \lesssim 1.0$. Compared with the cavitation patterns for the SS hydrofoil shown in figure 6, the maximum attached cavity length is slightly longer for the CF hydrofoil for $\sigma \gtrsim 0.7$. Moreover, the cavities extend all the way to the tip for $\sigma \lesssim 1.0$ for the CF hydrofoil, but only for $\sigma \lesssim 0.7$ for the SS hydrofoil. Tip vortex cavities can also be observed for $0.3 \lesssim \sigma \lesssim 0.7$ in panels (a–e): (a) $\sigma =0.3$; (b) $\sigma =0.4$; (c) $\sigma =0.5$; (d) $\sigma =0.60$; (e) $\sigma =0.7$; ( f) $\sigma =0.8$; (g) $\sigma =0.9$; (h) $\sigma =1.0$.

Figure 8. Variation of the measured (Exp, based on data presented by Smith et al. Reference Smith, Venning, Pearce, Young and Brandner2020b) and the modelled (Pre, obtained using (3.8)) normalized attached cavity length ($L_c/c$) with the cavitation number ($\sigma$). The measured values correspond to the maximum local attached cavity length normalized by the local chord at different normalized spanwise locations ($y/s$) and are shown in open circles and cyan filled squares for the SS and CF hydrofoils, respectively. The modelled values for the SS and CF hydrofoils, obtained using (3.8), are shown as a blue dashed line and a magenta dash–dotted line, respectively. The cavity length is slightly longer for the CF hydrofoil than the SS hydrofoil for $\sigma \gtrsim 0.7$ because the CF hydrofoil has a higher effective angle of incidence owing to flow-induced nose-up twist.

In general, cloud cavitation was observed on both hydrofoils for $0.3 \lesssim \sigma \lesssim 1.0$, and supercavitation for $\sigma \lesssim 0.3$. For $1.0 \lesssim \sigma \lesssim 1.2$, relatively stable partial leading edge sheet cavitation was observed for the SS hydrofoil, while unsteady cloud cavitation was observed for the CF hydrofoil. The earlier transition of the CF hydrofoil to cloud cavitation was primarily owing to the higher effective angle of incidence caused by the nose-up twist allowed by the greater twisting flexibility of the CF hydrofoil. It is important to note that while the imperfect surface finish of the CF hydrofoil triggered local bubble cavitation inception, it did not impact the global cavitation behaviour, particularly in the cloud and supercavitation regimes.

Figures 6(e–h) and 7(e–h) show Type II re-entrant jet cavity shedding on the SS and CF hydrofoils, respectively, for $0.7 \lesssim \sigma \lesssim 1.0$. Figures 6(b–d) and 7(b–d) show combined Type II re-entrant jet and Type I shock-wave-driven cavity shedding on the SS and CF hydrofoils, respectively, for $0.4 \lesssim \sigma \lesssim 0.6$. Figures 6(a) and 7(a) show Type I shock-wave-driven cavity shedding only at $\sigma = 0.3$. Relatively stable supercavitation was observed when $\sigma <0.3$ ($L_c/c\gtrsim 1.5$), but it is not shown in figures 6 and 7 to be succinct. Interested readers should refer to Parts I (Smith et al. Reference Smith, Venning, Pearce, Young and Brandner2020a) and II (Smith et al. Reference Smith, Venning, Pearce, Young and Brandner2020b) for more detailed images, including a breakdown of the Type I and Type II cavity shedding mechanics, frequencies and phase plots.

Figure 8 shows good general agreement between the predicted and measured maximum attached cavity length, which increases with decreasing $\sigma$. An inflection point is observed in the measured maximum attached cavity length ($L_c/c$) when $\sigma \approx 0.7$ or $L_c/c \approx 0.8$. Here, $\sigma \approx 0.7$ corresponds to when shock-wave-driven cavity shedding was first observed as the cavitation number was lowered for both hydrofoils. There was a reduction in the cavity length growth rate for $0.4 \lesssim \sigma \lesssim 0.6$, which corresponds to $L_{c}/c = 1.0$, at which the cavity closure interacts with flow from the pressure side of the hydrofoil (Smith et al. Reference Smith, Venning, Pearce, Young and Brandner2020a,Reference Smith, Venning, Pearce, Young and Brandnerb).

As shown in figures 6–8, the CF hydrofoil exhibited slightly longer cavities than the SS hydrofoil because of a higher nose-up twist for $L_c/c \lesssim 0.8$ or $\sigma \gtrsim 0.7$. The small difference between the SS and CF hydrofoil arose from the offset of the higher twisting flexibility of the CF hydrofoil by the reduced moment arm, as the mean EA of the CF hydrofoil was upstream of the midchord (i.e. $\bar {a}<0$, as shown in figure 2). The $L_c/c$ was nearly the same for both hydrofoils when $\sigma \lesssim 0.7$ because of the limited pitching moment in supercavitating flow. It can also be seen in figure 8 that $L_c/c$ differed the most between the SS and CF hydrofoils for the measurement points farthest away from the root, e.g. $y/s = 0.6~and~0.8$, where spanwise bending and twist deflection were the highest. Comparison of the cavitation patterns in figures 6(e–h) and 7(e–h) indicated that while the cavity extended across the entire span of the CF hydrofoil, including at the tip, for $\sigma \lesssim 1.0$, the cavity did not extend all the way to the tip of the SS hydrofoil until $\sigma \lesssim 0.7$. Another noteworthy observation from figures 6 and 7 is the presence of the tip vortex cavities for $0.3 \lesssim \sigma \lesssim 0.7$ for both hydrofoils.

4.3. Mean performance and histograms in cloud cavitation

The measured variation of the mean normal force ($C_N$) and mean pitch moment ($C_P$) coefficients about the midchord, and normalized CP from midchord ($e$) with cavitation number ($\sigma$) are shown as black open circles for the SS and CF hydrofoils in figure 9. The magenta lines in figure 9 correspond to the measured histograms in the form of probability density functions at each $\sigma$. In general, the mean $C_N$ and $C_P$ reduce as the cavity lengthens. The local maximum in $C_N$ occurs at $\sigma \approx 0.7$ or $L_c/c \approx 0.8$ because of the large partial cavity maximizing the virtual camber effect while being just far enough away from the hydrofoil trailing edge such that it does not significantly modify the flow (and hence pressure) on the pressure side of the hydrofoil. Note that $\sigma \approx 0.7$ or $L_c/c \approx 0.8$ is also where an inflection point can be observed on $L_c/c$ in figure 8. As $\sigma$ decreases from 0.7, the cavity lengthens to reach the hydrofoil trailing edge and beyond, which causes the pressure on the pressure side to lower and reach the vapour pressure at the hydrofoil trailing edge (as required by the Kutta condition when it is covered by the vaporous cavity). The lower net pressure difference leads to a reduction in $C_N$. However, as $\sigma$ increases from 0.7, $L_c/c$ decreases, which reduces the virtual camber effect, as well as the effect of the cavity on the suction side pressure distribution. Hence, $C_N$ gradually reduces and approaches the fully wetted value as $\sigma$ further increases from 0.7. Note that the reduction in $C_P$ occurs earlier (at a higher $\sigma$) because of the shift in CP towards the midchord ($e \rightarrow 0$) as $L_c/c \rightarrow 1$. The mean $e$ reaches a minimum near $\sigma \approx 0.6$, which corresponds to when $L_c/c \approx 1.0$ according to figure 8. For $\sigma <0.6$, the mean CP shifts back towards the quarter chord position because of increases in the suction side pressure caused by the re-entrant jet and shock-wave induced shedding. The shedding and breakdown of the low-pressure vaporous cavity shift upstream because shock-waves travel faster and further upstream than the re-entrant jet. This explains the trend for the mean value of $e$ and the sustained high amplitude of fluctuations in $e$ for $\sigma <0.6$.

Figure 9. Variation of the measured mean normal force ($C_N$) and mean pitch moment ($C_P$) coefficients about the midchord, and normalized centre of pressure (CP) from midchord ($e$) with cavitation number ($\sigma$) are shown as black open circles for the SS (a,c,e) and CF (b,d, f) hydrofoils. The magenta lines correspond to the measured histograms in the form of probability density functions at each $\sigma$. In general, the mean $C_N$ and $C_P$ reduces as $\sigma$ reduces, but the drop in $C_P$ occurs at a higher $\sigma$ than $C_N$ because of the reduction in $e$. The load fluctuations initially increase as the cavity lengthens with reduction in $\sigma$, and reach a maximum when the cavity approaches the hydrofoil trailing edge ($\sigma \approx 0.6$), which corresponds to when $e$ is the lowest (the CP is nearest to the midchord). For $\sigma <0.6$, the load fluctuations decrease as supercavitation develops with further reduction in $\sigma$, while $e$ increases to near the fully wetted value, as the re-entrant jet- and shock-wave-driven cavity shedding lead to a higher suction side pressure upstream of the trailing edge.

There is minimal load fluctuations for when $\sigma \gtrsim 1.0$ for the SS hydrofoil and $\sigma \gtrsim 1.1$ for the CF hydrofoil. Fluctuations begin at a higher $\sigma$ for the CF hydrofoil because of an earlier transition to cloud cavitation, consistent with experimental observations reported in Part II (Smith et al. Reference Smith, Venning, Pearce, Young and Brandner2020b) and with the slightly longer cavity observed in figure 8. The load fluctuations initially increase as the cavity lengthens with a reduction in $\sigma$, and reach a maximum when the cavity approaches the hydrofoil trailing edge ($\sigma \approx 0.6$). For $\sigma <0.6$, the load fluctuations decrease as supercavitation develops with a further reduction in $\sigma$. The variation of the amplitude of the fluctuating normal force coefficients is captured by (3.17), where the fluctuations decay to zero ($C_{Ro}^{\prime } \rightarrow 0$) in stable supercavitation as $\psi _e \rightarrow 0$ and in fully wetted flow as $\psi _e \rightarrow \infty$. The highest fluctuation in $C_N$ and $C_P$ is observed at $\sigma \approx 0.6$ or $L_c/c \approx 1$, which corresponds to $\psi _e \approx 2.8$ with $\alpha _e = 6^{\circ }$, consistent with the local maxima that can be obtained by taking the derivative of (3.17) with $\psi _e$.

The measured variation of the mean normalized tip bending deflection ($\delta /c$) and tip twist angle ($\theta$) with $\sigma$ are shown as black open circles for the SS and CF hydrofoils in figure 10. The magenta lines correspond to the measured histograms in the form of probability density functions at each $\sigma$. Comparison with figure 9 shows that, in general, the trend for $\delta$ follows $C_N$, and the trend for $\theta$ follows $C_P$, as the hydrofoils behave linear elastically. Both hydrofoils underwent slight bending towards the suction side. Compared with the CF hydrofoil, the SS hydrofoil experiences a much lower mean amplitude of tip bending and twisting deformations, which is expected because of the higher bending and twisting stiffness values ($K_s^{\delta \delta }$ and $K_s^{\theta \theta }$). In particular, the tip twist of the SS hydrofoil is near $0^{\circ }$, i.e. negligible. In fully wetted and partially cavitating flows, the CF hydrofoil undergoes nose-up twist, $\theta >0$ for $\sigma >0.75$ in figure 10. Here, $\theta$ reduces as $\sigma$ reduces to near 0.6 because of the reduction in $e$ (i.e. CP moving towards the midchord), and reaches a minimum at $\sigma \approx 0.6$ with $e \approx 0.1$. Additionally, $\theta <0$ for $0.4 \lesssim \sigma \lesssim 0.7$ because the CP is shifted to aft of the EA at the tip ($e < 0.12$), causing a nose-down twist. This is because the EA at the tip is noticeably forward of the midchord ($a(\bar {y}=1)=-0.12$, as shown in figure 2). Last, $\theta \approx 0$ for $\sigma \le 0.3$ because of the low value of $C_P$ in supercavitating flow.

Figure 10. Variation of the measured the mean normalized tip bending deflection ($\delta /c$) and tip twist angle ($\theta$) with cavitation number ($\sigma$) are shown as black open circles for the SS (a,c) and CF (b,d) hydrofoils. The magenta lines correspond to the measured histograms in the form of probability density functions at each $\sigma$. In general, the trend for $\delta$ follows $C_N$, and the trend for $\theta$ follows $C_P$, as the hydrofoils behave linear elastically. The SS hydrofoil experiences a much lower mean amplitude of tip deformations because of the higher stiffness compared with the CF hydrofoil. In fully wetted flow, the CF hydrofoil undergoes nose-up twist, but $\theta$ reduces as $\sigma$ reduces to near 0.6 because of the reduction in $e$ (i.e. CP moving towards the midchord). Here, $\theta <0$ for $0.3 \lesssim \sigma \lesssim 0.75$ because the CP is shifted to aft of the EA at the tip, which causes a slight nose-down twist. Additionally, $\theta \approx 0$ for $\sigma \le 0.3$ because of the low value of $C_P$ in supercavitating flow.

The measured histograms of $C_P$ and $e$ in figure 9, as well as $\theta$ in figure 10, exhibit increasing peakiness, or Kurtosis, as $\sigma$ decreases from 0.7, which is the cavitation number when shock-wave-driven cavity shedding was first observed. The results are consistent with the physical observations that shock-wave-driven cavity shedding is more coherent along the span, while re-entrant jet cloud cavity shedding exhibits more 3-D variations along the span. In addition, Type II subharmonic lock-in occurs on both hydrofoils at $\sigma \approx 0.7$, which may contribute to the large fluctuations in normal force.

The histograms of the normal force coefficient ($C_N$) and tip bending displacement ($\delta$) show similar behaviour to $C_P$ and $\theta$, respectively, except at $\sigma =0.4$ for the CF hydrofoil, where a bi-modal response is observed in figures 9 and 10. The bi-modal response is caused by the strong presence of Type I and Type II cavity shedding, and by lock-in of the Type I cavity shedding frequency with the nearest subharmonic of the system bending frequency of the CF hydrofoil. The subharmonic lock-in will be explained in more detail in § 4.5.

4.4. Steady-state FSI response in cavitating flow

Comparisons of the measured and predicted mean normal force ($C_N$) and mean pitch moment ($C_P$) coefficients about the midchord, as well as normalized tip bending deflection ($\delta /c$) and mean tip twist angle ($\theta$), for the SS and CF hydrofoils are shown in figure 11. The measured and predicted normalized position of the CP from the midchord, $e$, are shown in figure 12. The experimental data for the SS and CF hydrofoils are shown by blue open circles and magenta open squares, respectively. The predictions obtained using the 2-DOF steady-state FSI model presented in § 3.1 are shown as blue dashed lines and magenta dash–dotted lines for the SS and CF hydrofoils, respectively.

Figure 11. (a,c) Comparison of the measured and predicted mean normal force ($C_N$) and mean pitch moment ($C_P$) coefficients about the michord, and (b,d) normalized tip bending deflection ($\delta /c$) and mean tip twist angle ($\theta$), with cavitation number ($\sigma$). The experimental data for the SS and CF hydrofoils are shown by blue open circles and magenta open squares, respectively. Good agreement is observed between the predictions and measurements for $C_N$, $C_P$ and $\delta /c$ for both hydrofoils. There is good agreement on the trend of the $\theta$ between prediction and measurement, but the amplitude is under-predicted because the spanwise-averaged normalized elastic axis position from the midchord, $\bar {a}$, is used in the prediction.

Figure 12. Variation of the measured and fitted normalized distance of the CP from the midchord, $e$, with cavitation number ($\sigma$). The experimental data for the SS and CF hydrofoils are shown by blue open circles and magenta open squares, respectively. The values predicted by (3.9) for the SS and CF hydrofoils are shown as blue dashed lines and magenta dash–dotted lines, respectively. As $\sigma$ reduces toward 0.6, $e$ lowers as the CP moves toward the midchord as $L_c/c \rightarrow 1$. As $\sigma$ further reduces from 0.6 and $L_c/c>1$, the CP moves back toward the quarter chord.

In general, good agreement is observed between the predictions and measurements of the mean hydrodynamic loads and deformations. The SS hydrofoil underwent some bending deformation with negligible twist, with the maximum mean tip bending deflection ($\delta$) less than 5 % of the mean chord length ($c$). The CF hydrofoil underwent greater bending deflection, with the maximum value being approximately 11.4 % of the mean chord. The SS hydrofoil exhibited negligible twist. In fully wetted and partially cavitating flow, the CF hydrofoil exhibited nose-up twist at the tip, as the CP was upstream of the EA. The nose-up twist was responsible for the slightly higher values of $L_c/c$, $C_N$ and $C_P$ of the CF hydrofoil. The higher nose-up twist, which was maximum at the free tip, was also why the cavity extended across the entire span of the CF hydrofoil for $\sigma \lesssim 1.0$, but only for $\sigma \lesssim 0.7$ for the SS hydrofoil, as observed when comparing figures 6 and 7.

The measured tip twist of the CF hydrofoil was negative for $0.4 \lesssim \sigma \lesssim 0.7$, which corresponded to when $e < 0.12$ and $\epsilon =e+a<0$, i.e. the CP was aft of the EA at the tip. For $\sigma \lesssim 0.7$, the flow-induced twist became insignificant ($|\theta |<0.5^{\circ }$), because of the drop in $C_N$ and $C_P$ as supercavitation developed with further reduction in $\sigma$. Consequently, $C_N$, $C_P$ and $L_c/c$ were nearly identical for the CF and SS hydrofoils for $\sigma \lesssim 0.7$. The amplitude of $\theta$ was under-predicted for the CF hydrofoil, as the spanwise-averaged EA was used in the ROM, and $|\bar {a}|<|a(\bar {y}=1)|$. The sectional normal force distribution was maximum near the root and went to zero at the tip (because of the image effect created by the fixed boundary at the root and the force free condition at the tip). The change in effective incidence angle caused by the twist at the tip did not impact the integrated loads, as the sectional load at the tip was zero. However, the change in the effective angle of incidence by twist inboard of the tip had a non-negligible impact on the integrated hydrodynamic loads, and hence the spanwise averaged EA was used in the ROM predictions.

It is important to note that while values of $L_c/c$, $C_N$ and $C_P$ of the CF hydrofoil were slightly higher than those of the SS hydrofoil owing to the nose-up twist of the CF hydrofoil, the effects were dampened by the reduced eccentricity of the CF hydrofoil. Because the mean EA of the CF hydrofoil was forward of the midchord ($\bar {a} < 0$) in the outboard portion of the foil, $\epsilon$ was reduced, and the experienced $C_P$ was lower. This difference was caused by the challenges of manufacturing a small-scale CF hydrofoil as discussed in § 2.1. An ideal CF hydrofoil in which the carbon fibre layers extend to the trailing edge would have a greater difference in values of $L_c/c$, $C_N$ and $C_P$ compared with those of the SS hydrofoil, as the effects of reduced eccentricity caused by $a<0$ would not be present.

4.5. Dynamic FSI response in cavitating flow

To understand the dynamic FSI response in cavitating flow, the fluctuating normal force and moment coefficients, $C_N^{\prime }$ and $C_P^{\prime }$, are analysed using the wavelet synchrosqueezed transform (WSST) implemented in Matlab, which is based on the work of Thakur et al. (Reference Thakur, Brevdo, Fučkar and Wu2013). WSST is used instead of the continuous wavelet transform to minimize energy smearing. The minimization of energy smearing allows for a clearer view of the interactions between the Type I and Type II cavity shedding frequencies, $\,f_{c1}$ and $\,f_{c2}$, and the cavity-induced modulation of the system bending frequency, $\hat {f}_{n1}$.

The measured time–frequency spectra obtained using the WSST of the fluctuating normal force coefficient ($C_N^{\prime }$) of the SS and CF hydrofoils are shown in figures 13 and 14, respectively. To facilitate interpretation of the graphs, markers are added on the right y-axis to indicate the predicted values for $\,f_{c1}$, $\,f_{c2}$ and $\hat {f}_{n1}$ given by (3.18), (3.19) and (3.22), respectively. The filled and open triangle markers on the right axis indicate the predicted Type I and Type II cavity shedding frequencies ($\,f_{c1}$ and $\,f_{c2}$) as given by (3.18) and (3.19), respectively. The magenta crosses indicate the range of the predicted system bending frequency as given by (3.20)–(3.22). Red dash markers are added on the left $y$-axis to indicate the predicted heterodyne frequencies, $\,f_{c1}+f_{c2}$, $\,f_{c2}-f_{c1}$, $2f_{c1}$ and $2f_{c2}$, for the spectra of $0.3 \lesssim \sigma < 0.7$, where both Type I and Type II cavity shedding are present.

Figure 13. Measured time–frequency spectra of the fluctuating normal force coefficient ($C_N^{\prime }$) of the SS hydrofoil at selected values of $\sigma$. The coloured contour in the plot shows energy concentration in dB ranging from $-20$ (yellow) to $-100$ (blue). The filled and open triangle markers on the right axis indicate the predicted Type I and Type II cavity shedding frequencies ($\,f_{c1}$ and $\,f_{c2}$), respectively. The magenta crosses indicate the range of the predicted system bending frequency. The red dash markers on the left axis indicate the predicted heterodyne frequencies ($\,f_{c1}+f_{c2}$, $\,f_{c2}-f_{c1}$, $2f_{c1}$ and $2f_{c2}$). (a) $\sigma= 0.3$; (b) $\sigma= 0.4$; (c) $\sigma= 0.5$; (d) $\sigma= 0.6$; (e) $\sigma= 0.7$; ( f) $\sigma = 0.8$; (g) $\sigma= 0.9$; (h) $\sigma= 1.0$.

Figure 14. Measured time–frequency spectra of the fluctuating normal force coefficient ($C_N^{\prime }$) of the CF hydrofoil at selected values of $\sigma$. The coloured contour in the plot shows energy concentration in dB ranging from $-20$ (yellow) to $-100$ (blue). The filled and open triangle markers on the right axis indicate the predicted Type I and Type II cavity shedding frequencies ($\,f_{c1}$ and $\,f_{c2}$), respectively. The magenta crosses indicate the range of the predicted system bending frequency. The red dash markers on the left axis indicate the predicted heterodyne frequencies ($\,f_{c1}+f_{c2}$, $\,f_{c2}-f_{c1}$, $2f_{c1}$ and $2f_{c2}$). (a) $\sigma= 0.3$; (b) $\sigma= 0.4$; (c) $\sigma = 0.5$; (d) $\sigma = 0.6$; (e) $\sigma = 0.7$; ( f) $\sigma= 0.8$; (g) $\sigma= 0.9$; (h) $\sigma= 1.0$.

As noted in (3.18), Type I cavity shedding appears for $0.9 \le \psi _e \le 3.3$ only, which corresponds to $0.3 \lesssim \sigma < 0.7$ (or $0.8 < L_c/c \lesssim 1.3$), where energy concentrations at $\,f_{c1}$ can be observed for both hydrofoils. Type I cavity shedding is absent for $\sigma \ge 0.7$, as the void fraction is too low, and the speed of sound is too high for shock-waves to develop. In contrast, energy concentrations at $\,f_{c2}$ can be observed for the full range of $\sigma$ shown in figures 13 and 14. The two cavity shedding mechanisms can occur simultaneously, or at different time instances, which leads to a phase difference that depends on the interaction between the two driving frequencies. The two cavity shedding frequencies also interact and form heterodyne frequencies. Hence, energy concentration can be observed at $\,f_{c2}-f_{c1}$, $2f_{c1}$, $\,f_{c2}+f_{c1}$ and $2f_{c2}$ for $0.3 \lesssim \sigma < 0.7$. The heterodyne frequencies are only observed intermittently because the higher order effects are most significant when the two cavity shedding frequencies are in phase.

As explained in § 3.2, periodic changes in the added mass of the liquid–vapour mixture caused by unsteady cavity shedding lead to time and frequency modulation of the system bending frequency, which can be observed in the fluctuation of energy concentrations near $\,f_{n1}$ in figures 13 and 14. Note that the system bending frequency was excited by hydrodynamic load variations caused by unsteady cavity shedding only, as the hydrofoils were subject to uniform flow and the test section turbulence level was minimal, at levels of approximately 0.6 % (Brandner et al. Reference Brandner, Lecoffre and Walker2007). Hence, energy concentrations near $\,f_{n1}$ were present only if $\,f_{c2}$ or $\,f_{c2}+f_{c1}$ was near $\,f_{n1}$, and if the intensity of the cavity fluctuating force was sufficiently large to excite the system bending mode. According to (3.17) and supported by the histograms shown in figures 9 and 10, the amplitude of the cavity excitation force was the highest at $\sigma \approx 0.6 - 0.7$ when $L _c/c \approx 1$ for $\alpha _e=6^{\circ }$, and it decayed to near zero for $\sigma >1$ (because only a small partial cavity was present or the flow was fully wetted) and $\sigma <0.2$ (because the flow was in the stable supercavitation regime). When the cavity shedding frequencies were far from the system bending frequency or the amplitude of the excitation force was low, e.g. $\sigma =0.3$, the system bending mode was not excited and no energy concentration was observed near $\,f_{n1}$. As the cavitation number $\sigma$ increased from 0.3 to 0.7, the Type II cavity shedding frequency $\,f_{c2}$ increased and gradually approached $\,f_{n1}$, and the energy concentration at $\,f_{n1}$ increased until $\sigma =0.7$. As $\sigma$ continued to increase past 0.7, the amplitude of the cavity fluctuating force decreased, and the energy concentration of the system bending mode also decreased. As such, in figure 13, the energy concentration near $\,f_{n1}$ was largest at $\sigma =0.7$, where $\,f_{c2}$ was nearest to $\,f_{n1}$. Similarly, in figure 14, the energy concentration near $\,f_{n1}$ was largest at $\sigma =0.6$, where $\,f_{c1}+f_{c2}$ was nearest to the system bending mode. Owing to the high stiffness of the model-scale hydrofoils, $\,f_{n1}$ was significantly higher than both $\,f_{c1}$ and $\,f_{c2}$ for most cavitation numbers, but at $\sigma \approx 0.9$, $\,f_{c2} \approx f_{n1}$ for both the SS and CF hydrofoils. However, because of the low intensity of the cavity fluctuating force and the high fluid damping (as shown in figure 4), primary lock-in did not occur in either hydrofoil, as can be seen in figures 13 and 14. The time–frequency spectra at $\sigma =1.0$ resembled broadband noise with very little participation from $\,f_{c2}$, as the cavity fluctuating force was low. The low cavity fluctuating force was attributed to the short cavity length $L_c$, as shown in figures 6, 7 and 8.

Interactions between the cavity shedding frequencies and system bending frequency were observed through enhancement of the energy intensity when $\,f_{n1}\approx 2f_{c2}$ at $\sigma =0.7$ for the SS hydrofoil, and by mingling of $\,f_{n1}\approx 2f_{c2}$ and $\,f_{c2}$ at $\sigma =0.6$ for the CF hydrofoil. At $\sigma =0.4$ for the CF hydrofoil, most of the energy was concentrated at $\,f_{c1}$, as the Type I cavity shedding frequency was locked into the nearest subharmonic of the system bending frequency, which happened to be $\,f_{n1}/4$.

While the FSI in cavitating flow is not a stationary process, it is still worthwhile to compare the PSDs of $C_N^{\prime }$ and $\delta ^{\prime }/c$ to understand the differences in the FSI between the SS and CF hydrofoils. The PSDs are shown in figures 15 and 16. The open blue circle and open magenta square indicate the location of the highest peak. The vertical blue dashed and magenta dash–dotted lines indicate the predicted mean system bending frequency ($\,f_{n1}$) of the SS and CF hydrofoils, respectively. The blue and magenta crosses on the top axis indicate the predicted Type I and Type II cavity shedding frequencies, $\,f_{c1}$ and $\,f_{c2}$, for the SS and CF hydrofoils, respectively. In general, compared with the SS hydrofoil, the amplitude of the fluctuating normal force coefficient, $C_N^{\prime }$, of the CF hydrofoil is lower (except at $\sigma =0.4$ owing to subharmonic lock-in), while the amplitude of the fluctuating tip bending deformation, $\delta ^{\prime }/c$, of the CF hydrofoil is higher. In addition, the location of the peaks for $f<40$ Hz for $\sigma <0.7$ are approximately the same between the SS and CF hydrofoils, which suggests similar cavity shedding frequencies for the SS and CF hydrofoils when the mean tip twist is small ($\theta \approx 0^{\circ }$ for the SS hydrofoil, while $|\theta |<0.5^{\circ }$ for the CF hydrofoil, as shown in figure 11). In addition, the Type I shedding frequency near $f=10{-}12$ Hz is the same for both hydrofoils. However, for $\sigma >0.7$, $\theta > 0.5^{\circ }$ for the CF hydrofoil, which increases $\alpha _e$, leading to a longer cavity and lower $\,f_{c2}$ compared with the SS hydrofoil. As such, the Type II cavity shedding frequency of the CF hydrofoil is consistently lower than the SS hydrofoil for $\sigma >0.7$, and the difference increases with increasing $\sigma$, where nose-up twisting becomes more apparent.

Figure 15. Measured $C_N^{\prime }$ PSD for the SS and CF hydrofoils at different values of $\sigma$ shown as thick blue and thin magenta lines, respectively. The open blue circle and open magenta square indicate the location of the peak with the highest magnitude. The vertical blue dotted and magenta dash–dotted lines indicate the predicted mean system bending frequency ($\,f_{n1}$) for the SS and CF hydrofoils, respectively. The blue and magenta crosses on the top axis indicate the predicted Type I and Type II cavity shedding frequencies, $\,f_{c1}$ and $\,f_{c2}$, for the SS and CF hydrofoils, respectively. The blue and magenta vertical dashes on the top axis indicate the predicted heterodyne frequencies, $\,f_{c2}-f_{c1}$, $2f_{c1}$, $\,f_{c2}+f_{c1}$ and $2f_{c2}$, for the SS and CF hydrofoils, respectively. The location of the peaks for $f<40$ Hz for $\sigma <0.7$ are approximately the same between the SS and CF hydrofoils, which suggests similar cavity shedding frequencies for the SS and CF hydrofoils when the mean tip twist is small. When $\sigma \ge 0.7$, the CF hydrofoil with greater nose-up twist has a slightly longer cavity length and lower Type II cavity shedding frequency compared with the SS hydrofoil. (a) $\sigma= 0.3$; (b) $\sigma= 0.4$; (c) $\sigma= 0.5$; (d) $\sigma = 0.6$; (e) $\sigma= 0.7$; ( f) $\sigma= 0.8$; (g) $\sigma= 0.9$; (h) $\sigma= 1.0.$

Figure 16. Measured $\delta ^{\prime }/c$ PSD for the SS and CF hydrofoils at different values of $\sigma$ shown as thick blue and thin magenta lines, respectively. The open blue circle and open magenta square indicate the location of the peak with the highest magnitude. The vertical blue dashed and magenta dash–dotted lines indicate the predicted mean system bending frequency ($\,f_{n1}$) for the SS and CF hydrofoils, respectively. The blue and magenta crosses on the top axis indicate the predicted Type I and Type II cavity shedding frequencies, $\,f_{c1}$ and $\,f_{c2}$, for the SS and CF hydrofoils, respectively. The blue and magenta dashes on the top axis indicate the predicted heterodyne frequencies, $\,f_{c2}-f_{c1}$, $2f_{c1}$, $\,f_{c2}+f_{c1}$ and $2f_{c2}$, for the SS and CF hydrofoils, respectively. The Type II cavity shedding frequency of the CF hydrofoil is consistently lower than the SS hydrofoil for $\sigma >0.7$, and the difference increases with increasing $\sigma$, where nose-up twisting becomes more apparent. (a) $\sigma= 0.3$; (b) $\sigma= 0.4$; (c) $\sigma= 0.5$; (d) $\sigma$ = 0.6; (e) $\sigma= 0.7;$ ( f) $\sigma= 0.8$; (g) $\sigma= 0.9$; (h) $\sigma= 1.0.$

Another noteworthy observation of the results shown in figures 15 and 16 is that peaks near $\,f_{n1}$ can only be observed for $0.7 \le \sigma \le 0.9$; the amplitude of the peaks increases as $\,f_{c2}$ approaches $\,f_{n1}$, which arises from dynamic load amplification near resonance. It should be noted that the peaks near $\,f_{n1}$ are smeared in the PSD, as the instantaneous system bending natural frequency (shown in figures 13 and 14) oscillates in time owing to cavity-induced modulation of the fluid added mass. The presence of heterodyne frequencies can be seen for $0.3 \lesssim \sigma < 0.7$, but they tend to be smeared as well because of the intermittent nature of the heterodyne frequencies, which depends on the interactions between $\,f_{c1}$ and $\,f_{c2}$.

Comparisons between figures 15 and 16 show that the frequencies of the highest peaks for $C_N^{\prime }$ and $\delta ^{\prime }/c$ are different for some values of $\sigma$. To understand the differences, the variation of the measured amplitude and frequency at the highest peaks of $C_N^{\prime }$ and $\delta ^{\prime }/c$ spectra (the locations identified by the open blue circles and open magenta squares in figures 15 and 16) are plotted against $\sigma$ in figures 17 and 18, respectively. In each figure, the amplitude of the highest peak is shown in panel (a), the peak frequency ($\,f_p$) normalized by the mean system bending frequency, $\,f_{n1}$, is shown in panel (b), and the peak frequency normalized by $\,f_{c2}$ and $\,f_{c_1}$ are shown on the left and right y-axis, respectively, in panel (c).

Figure 17. Variation of the measured amplitude and frequency at the highest peak of the PSD of fluctuating normal force coefficient ($C_N^{\prime }$) for the SS and CF hydrofoils: (a) amplitude at the peak of the $C_N^{\prime }$ spectra, $C_{N,p}^{\prime }$; (b) peak frequency ($\,f_p$) normalized by the predicted system bending frequency, $\,f_{n1}$; (c) peak frequency normalized by the predicted Type II cavity shedding frequency ($\,f_{c2}$) on the left y-axis, and by the predicted Type I cavity shedding frequency ($\,f_{c1}$) on the right y-axis. The results suggest Type II subharmonic lock-in at $\sigma ~\approx 0.7$ for both hydrofoils, and Type I subharmonic lock-in at $\sigma ~\approx 0.4$ for the CF hydrofoil.

Figure 18. Variation of the measured amplitude and frequency at the highest peak of the PSD of the fluctuating normalized tip bending fluctuations ($\delta ^{\prime }/c$) for the SS and CF hydrofoils: (a) amplitude at the peak of the $\delta ^{\prime }$ spectra, $\delta _{p}^{\prime }/c$; (b) peak frequency ($\,f_p$) normalized by the predicted system bending frequency, $\,f_{n1}$; (c) peak frequency normalized by the predicted Type II cavity shedding frequency ($\,f_{c2}$) on the left $y$-axis, and by the predicted Type I cavity shedding frequency ($\,f_{c1}$) on the right $y$-axis. The results suggest Type II subharmonic lock-in at $\sigma \approx 0.7$ for both hydrofoils, and Type I subharmonic lock-in at $\sigma \approx 0.4$ for the CF hydrofoil.

For the SS hydrofoil, the local peak in $C_{N,p}^{\prime }$ between $\sigma =0.70 - 0.75$ is caused by lock-in of the Type II cavity shedding frequency with the subharmonic of the system bending frequency. The peak frequency for the SS hydrofoil is $\,f_p=f_{c2}=f_{n1}/2$ for $C_N^{\prime }$ and $\,f_p=f_{n1}=2f_{c2}$ for $\delta ^{\prime }/c$.

For the CF hydrofoil, the most prominent peak in $C_{N,p}^{\prime }$ and $\delta ^{\prime }/c$ at $\sigma \approx 0.4$ is caused by lock-in of the Type I cavity shedding frequency with the nearest subharmonic of $\,f_{n1}$, which is $\,f_{n1}/4$, as shown in figures 17 and 18. In other words, the peak frequency for the CF hydrofoil is $\,f_p=f_{c1}=f_{n1}/4$ for $C_N^{\prime }$ and $\delta ^{\prime }/c$. A smaller local peak in $C_{N,p}^{\prime }$ can also be observed at $\sigma \approx 0.70$ for the CF hydrofoil in figures 17 and 18, where $\,f_p=f_{n1}=3f_{c2}/2$, which suggests secondary Type II subharmonic lock-in.

An overall perspective of the dynamic response may be better visualized in the combined measured spectrograms of the normalized tip bending fluctuations for all cavitation numbers, as shown in figure 19. Energy concentrations can be observed along the predicted dash–dotted line for $\,f_{c1}$, dashed line for $\,f_{c2}$ and in between the dotted lines indicating the maximum and minimum values for the modulating system bending frequency, $\hat {f}_{n1}$. Higher energy concentrations (darker spots), which indicate increased deformations, can be observed at $\,f_{c2}$ between $\sigma \approx 0.70 - 0.75$, where the Type II cavity shedding frequency locks in with the subharmonic of the system bending frequency for the SS hydrofoil. For the CF hydrofoil, darker spots can be observed at $\,f_{c1}$ near $\sigma \approx 0.40$, where the Type I cavity shedding frequency locks in with the subharmonic of $\,f_{n1}$, and at $\,f_{c2}$ between $\sigma \approx 0.70 - 0.75$, where the Type II cavity shedding frequency locks in with the subharmonic of $\,f_{n1}$.

Figure 19. Comparison of the measured spectrograms of the normalized tip bending fluctuations ($\delta ^{\prime }/c$) for the SS (a) and CF (b) hydrofoils. The dash–dotted and dashed lines indicate the predicted Type I ($\,f_{c1}$) and Type II ($\,f_{c2}$) cavity shedding frequencies, respectively. The solid and dotted lines indicate the mean and range of the predicted system bending frequency ($\,f_{n1}$). Dark spots indicating high energy concentration can be observed at $\sigma ~\approx 0.7$ and $f \approx f_{c2}$ for both hydrofoils owing to Type II subharmonic lock-in, and at $\sigma ~\approx 0.4$ and $f \approx f_{c1}$ for the CF hydrofoil owing to Type I subharmonic lock-in.

Figure 19 indicates that $\,f_{c2}$ matches with $\,f_{n1}$ at $\sigma \approx 0.95$ for the SS hydrofoil and at $\sigma \approx 0.85$ for the CF hydrofoil. However, as shown in figures 15–19, no significant amplification can be observed at the frequency where primary lock-in is expected. The lack of amplification is caused by a limited cavity length when $\sigma \ge 0.85$ ($L_c/c \le 0.5$ according to figure 8), which yields low intensity for the cavity excitation force ($F_{Ro}^{\prime }$, as defined in (3.16) and (3.17)).

The predicted spectrograms of the normalized tip bending fluctuations ($\delta ^{\prime }/c$) for the SS and CF hydrofoils obtained using the 1-DOF parametric oscillator model described in § 3.2 are shown in figure 20. Comparison with figure 19 show good general agreement between predictions and measurements, but the 1-DOF model is not able to predict the increase in vibration caused by lock-in of the cavity shedding frequencies with the subharmonic of system bending frequency because of the use of the simplified linear damping model.

Figure 20. Comparison of the predicted spectrograms of the normalized tip bending fluctuations ($\delta ^{\prime }/c$) for the SS (a) and CF (b) hydrofoils. The dash–dotted and dashed lines indicate the predicted Type I ($\,f_{c1}$) and Type II ($\,f_{c2}$) cavity shedding frequencies, respectively. The solid and dotted lines indicate the mean and range of variation of the predicted system bending frequency ($\,f_{n1}$). Good general agreement is observed when compared with figure 19, but the prediction is not able to capture the dynamic vibration amplification arising from subharmonic lock-in.

To better understand the differences between the measurements and predictions, the measured (Exp) and predicted (Pre) time–frequency spectra of the normalized tip bending fluctuations ($\delta ^{\prime }/c$) at selected cavitation numbers for the SS and CF hydrofoils are shown in figures 21–24. Four sets of plots are shown in each figure, where each set consists of the time history, the power spectrum (PS), and the time–frequency spectrum (obtained using WSST) of $\delta ^{\prime }/c$. To facilitate interpretation of the graphs, blue open triangles and blue filled triangles are used to mark the predicted Type I ($\,f_{c1}$) and Type II ($\,f_{c2}$) cavity shedding frequencies, respectively, in the power spectra and in the time–frequency spectra. Magenta crosses are used to indicate the range of variation of the system bending natural frequency, $\,f_{n1}$. In addition, the heterodyne frequencies ($\,f_{c2} \pm f_{c1}$, $2f_{c1}$ and $2f_{c2}$), caused by mixing of $\,f_{c1}$ and $\,f_{c2}$ when both cavity shedding mechanisms are present, are indicated by the green dash–dotted lines on the left axis of the power spectra.

Figure 21. Comparison of the measured (Exp, a,c) and predicted (Pre, b,d) time and frequency spectra of the normalized tip bending fluctuations ($\delta ^{\prime }/c$) at $\sigma \approx 0.4$ for the SS (a,b) and CF (b,d) hydrofoils. The coloured time frequency contour in the plot shows energy concentration in dB ranging from $-20$ (yellow) to $-100$ (blue). The markers in the power spectra (PS) and in the time–frequency spectra indicate the predicted Type I and Type II cavity shedding frequencies ($\,f_{c1}$ and $\,f_{c2}$), as well as the range of the predicted system natural frequency ($\,f_{n1}$). In addition, the predicted heterodyne frequencies caused by mixing of $\,f_{c1}$ and $\,f_{c2}$ are indicated by the green dash–dotted lines in the power spectra. Good general agreement is observed between the predictions and measurements, except for the concentration of energy at $\,f_{c1}$ owing to Type I subharmonic lock-in for the CF hydrofoil. Note that the amplitude of the tip bending fluctuations are higher for the CF hydrofoil compared with the SS hydrofoil.

Good general agreement can be observed between predictions and measurements of the frequency and amplitude of $\delta ^{\prime }/c$ in figures 21–24. The fluctuating frequency spectra of both hydrofoils exhibit peaks at the Type II re-entrant jet-driven frequency ($\,f_{c2}$) for $0.3 \lesssim \sigma \lesssim 1.0$ and the Type I shock-wave-driven cavity shedding frequency ($\,f_{c1}$) for $0.2 \lesssim \sigma \lesssim 0.7$. Sporadic energy concentration can also be observed around the fundamental natural frequency, $\,f_{n1}$, which modulates owing to unsteady cavity shedding. When both cavity shedding mechanisms are present, the time–frequency spectra indicate transient modulations between $\,f_{c2}$ and $\,f_{c1}$, as the two frequencies are often not in phase and occur at different instances in time.

In the measured response of the CF hydrofoil, minor peaks can also be observed at the heterodyne frequencies caused by mixing of the two cavity shedding frequencies, as shown in figures 21 and 22 for $\sigma \approx 0.4$ and $\sigma \approx 0.5$, respectively. The heterodyne frequencies can also be observed on the PSD of the CF hydrofoil in figures 15 and 16 for $0.3 \lesssim \sigma < 0.7$. The presence of the heterodyne frequencies suggests that the hydrofoil acts as a nonlinear oscillator that results in the mixing of the frequencies. The heterodyne frequencies were not captured by the 1-DOF parametric oscillator predictions, likely owing to the high and linear fluid damping assumed in the model.

Figure 22. Comparison of the measured (Exp, a,c) and predicted (Pre, b,d) time and frequency spectra of the normalized tip bending fluctuations ($\delta ^{\prime }/c$) at $\sigma \approx 0.5$ for the SS (a,b) and CF (c,d) hydrofoils. The coloured time frequency contour in the plot shows energy concentration in dB ranging from $-20$ (yellow) to $-100$ (blue). The markers in the power spectra (PS) and in the time–frequency spectra indicate the predicted Type I and Type II cavity shedding frequencies ($\,f_{c1}$ and $\,f_{c2}$), as well as the range of the predicted system natural frequency ($\,f_{n1}$). In addition, the predicted heterodyne frequencies caused by mixing of $\,f_{c1}$ and $\,f_{c2}$ are indicated by the green dash–dotted lines in the power spectra. Good general agreement is observed between the predictions and measurements. The bending fluctuations are higher for the CF hydrofoil compared with the SS hydrofoil.

As shown in figures 19–24, the 1-DOF parametric oscillator model is able to predict the general time–frequency response of both the SS and CF hydrofoils, which includes the general higher amplitude of bending fluctuations for the CF hydrofoil. However, the model failed to predict the Type I subharmonic lock-in of the CF hydrofoil at $\sigma =0.4$ in figure 21 and the Type II subharmonic lock-in of the SS hydrofoil at $\sigma =0.7$ in figure 23.

Figure 23. Comparison of the measured (Exp, a,c) and predicted (Pre, b,d) time and frequency spectra of the normalized tip bending fluctuations ($\delta ^{\prime }/c$) at $\sigma =0.7$ for the SS (a,b) and CF (c,d) hydrofoils. The coloured time frequency contour in the plot shows energy concentration in dB ranging from $-20$ (yellow) to $-100$ (blue). The markers in the power spectra (PS) and in the time–frequency spectra indicate the predicted Type I and Type II cavity shedding frequencies ($\,f_{c1}$ and $\,f_{c2}$), as well as the range of the predicted system natural frequency ($\,f_{n1}$). Although there is general agreement on the measured and predicted amplitude, the prediction is not able to capture the higher energy concentration at $\,f_{n1}$ owing to Type II subharmonic lock-in.

Figure 24. Comparison of the measured (Exp, a,c) and predicted (Pre, b,d) time and frequency spectra of the normalized tip bending fluctuations ($\delta ^{\prime }/c$) at $\sigma =0.9$ for the SS (a,b) and CF (c,d) hydrofoils. The coloured time frequency contour in the plot shows energy concentration in dB ranging from $-20$ (yellow) to $-100$ (blue). The markers in the power spectra (PS) and in the time–frequency spectra indicate the predicted Type I and Type II cavity shedding frequencies ($\,f_{c1}$ and $\,f_{c2}$), as well as the range of the predicted system natural frequency ($\,f_{n1}$). Good general agreement is observed between the predictions of measurements. As indicated by the proximity of $\,f_{n1}$ to $\,f_{c2}$, this case should be near primary Type II lock-in for both hydrofoils. However, the amplitude of the bending fluctuations are low because of the low amplitude of the cavity excitation force with the small cavity as observed in figures 6(g), 7(g) and 8.

Comparisons of the measured and predicted statistics of $C_N^{\prime }$ and $\delta ^{\prime }/c$ as a function of $\sigma$ for the SS and CF hydrofoils are shown in figure 25. Here, $C_{N,10}^{\prime }$ and $\delta _{10}^{\prime }/c$ are the averaged amplitude of the highest 10 % of the fluctuating normal force coefficient and fluctuating normalized tip bending deflection, respectively, $C_{N,SD}^{\prime }$ and $\delta _{SD}^{\prime }/c$ are the standard deviation of the fluctuating normal force coefficient and fluctuating normalized tip bending deflection, respectively, and $C_{Ro}^{\prime }$ is the amplitude of the fluctuating normal force coefficient for an equivalent rigid hydrofoil given by (3.17). The results show that the general response of the SS hydrofoil is well predicted. However, in general, the 1-DOF parametric oscillator model over-predicted the amplitude of $C_N^{\prime }$ and $\delta ^{\prime }/c$ for the CF hydrofoil.

Figure 25. Variation of the measured (a,c) and predicted (b,d) statistics of the fluctuating normal force coefficient ($C_N^{\prime }$) and normalized tip bending fluctuation ($\delta ^{\prime }/c$) with cavitation number ($\sigma$) for the SS and CF hydrofoils. Here, $C_{N,10}^{\prime }$ and $\delta _{10}^{\prime }/c$ are the average amplitude of the highest 10 % of the fluctuating normal force coefficient and fluctuating normalized tip bending deflection, respectively, and $C_{N,SD}^{\prime }$ and $\delta _{SD}^{\prime }/c$ are the standard deviation of the fluctuating normal force coefficient and fluctuating normalized tip bending deflection, respectively. Also shown in panels (a,b) are the modelled amplitude of the fluctuating normal force coefficient for an equivalent rigid hydrofoil ($C_{Ro}^{\prime }$ given in (3.17)). The measured amplitude of $C_{N}^{\prime }$ is generally lower for the CF hydrofoil than the SS hydrofoil, while the amplitude of $\delta ^{\prime }/c$ is generally higher for the CF hydrofoil, because flow kinetic energy is being diverted to excite the structure and hence the fluid forces experienced by the flexible hydrofoil are less than those experienced by a rigid or stiff hydrofoil. An exception to this general trend is observed at $\sigma = 0.4$, where strong Type I subharmonic lock-in develops for the CF hydrofoil, which allows $C_N^{\prime }$ of the CF hydrofoil to surpass that of the SS hydrofoil.

The measured statistics shown in figure 25(a,c) indicate that the amplitude of $C_{N}^{\prime }$ is generally lower for the CF hydrofoil (except at $\sigma =0.4$ owing to dynamic load amplification caused by subharmonic lock-in) than the SS hydrofoil, while the amplitude of $\delta ^{\prime }/c$ is generally higher for the CF hydrofoil, which is consistent with observations of the PSDs shown in figures 15 and 16. The higher $\delta ^{\prime }/c$ is caused by the higher flexibility of the CF hydrofoil. The general lower $C_{N}^{\prime }$ of the CF hydrofoil is caused by the flow energy being diverted to vibrate the hydrofoil. Re-arranging (3.13) and (3.14) yields $F_R^{\prime }=F_N^{\prime }+( \hat {M}_f\ddot {\delta ^{\prime }}+C_f\dot {\delta ^{\prime }}+K_f{\delta ^{\prime }} )$. Because $\hat {M}_f$ and $C_f$ are positive, and $K_f$ is small in comparison (for the range of velocities considered), the amplitude of the fluctuation normal force for an equivalent rigid hydrofoil ($F_R^{\prime }=C_R^{\prime } qsc$) should be higher than that for the flexible hydrofoil ($F_N^{\prime }=C_N^{\prime } qsc$), and hence $|C_R^{\prime }|>|C_N^{\prime }|$. This is the reason for the dynamic load alleviation of the CF hydrofoil compared with the SS hydrofoil. Physically speaking, flow kinetic energy is being diverted to excite the structure and hence the fluid forces experienced by the flexible hydrofoil is less than that experienced by a rigid or stiff hydrofoil. Hence, the difference between $C_N^{\prime }$ of the CF and SS hydrofoils increases as $\delta ^{\prime }/c$ increases. An exception to this general trend is observed at $\sigma = 0.4$, where strong Type I subharmonic lock-in develops for the CF hydrofoil. During lock-in, the fluid damping decreases, leading to higher vibrations and hydrodynamic loads, which allows $C_N^{\prime }$ of the CF hydrofoil to surpass that of the SS hydrofoil.

The 1-DOF parametric oscillator model failed to predict the dynamic load amplification caused by subharmonic lock-in, as seen by the absence of peaks at $\sigma = 0.4$ in figure 25(b,d). This observation agrees with the observations made in comparing figures 19 and 20. Both of these failures in the model are likely caused by the assumption of simple linear damping. Subharmonic lock-in is highly sensitive to the form and the value of the damping (Náprstek & Fishcer Reference Náprstek and Fishcer2019), so modelling the damping incorrectly can result in the failure to capture subharmonic lock-in.

Although the modelled fluid damping coefficient is quite high for the $\sigma$ range of interest, which ranges between 25 % and 45 % as shown in figure 4, the model still over-predicts $\delta _{SD}^{\prime }/c$ for the CF hydrofoil. The model also slightly over-predicts the amplitude of $C_N^{\prime }$ for the CF hydrofoil. This suggest that a nonlinear damping component proportional to $(\delta ^{\prime })^{2}$ should be added to the fluid damping model to account for the increase in energy dissipation with higher flow-induced vibrations and the resulting dynamic load alleviation observed for $C_{N}^{\prime }$.