3.4.1 Shaker excitation and drive-point measurements

A linear shaker motor (model 2007E ‘mini-shaker’, manufactured by The Modal Shop^®) was attached to the suction side of the hydrofoil above the still waterline to provide a harmonic excitation force at a commanded frequency. Forces and accelerations at the drive point were measured with a PCB Piezotronics^® impedance head (model 288D01). Excitation signals were generated as sinusoid waveforms from a Stanford Research Systems^® SR830 lock-in amplifier, phase locked to a TTL pulse train generated by the data collection computer. The excitation signal was a discretely stepped logarithmic sweep, wherein each commanded frequency was held constant for an integer number of cycles $N_{C}$, as shown in figure 5. Sweeps were conducted over a frequency band sufficient to excite the first five resonant modes (between 2 Hz and 140 Hz or more) with frequency step sizes of 0.1 Hz to 0.3 Hz. Convergence testing demonstrated that modal parameters did not vary strongly with the number of periods for $N_{C}\geqslant 5$, and all reported trials were conducted with $N_{C}\geqslant 10$.

During vibration testing, runs containing non-stationary responses, such as formation or elimination of a ventilated cavity, were voided and repeated until a complete frequency sweep was attained in an unchanging flow regime. Other sources of non-stationarity, such as inflow fluctuations or partial cavity shedding were an unavoidable source of error. In particular, fluctuations in cross-flow angle were observed to occur at frequencies below 10 Hz, introducing some spurious spectral content, discussed further in § 4.2. A zero-phase-shift digital high-pass filter was used during post-processing to remove mean and slowly varying forces and deflections below 2 Hz. As mentioned in Part 1, some unavoidable sensor drift was observed, particularly in the shape-sensing spars. Regular bias measurements and filtering ensured that the slow drift had no effect upon the higher-frequency dynamics discussed in this work.

Figure 5. Typical shaker motor excitation signal with $N_{C}=25$ complete periods at each frequency.

4.2.1 Dry conditions

The magnitude of the compliance FRF, averaged across all degrees of freedom and decomposed into the contributions from each mode (modal participation factors), is shown in figure 6. Modes 1, 2, 3 and 5 are observed experimentally. The modal frequencies predicted with the FEA model are overlaid and labelled as vertical lines. At specific frequencies, components of the mounting structure or facility resonated, creating artificial peaks in the measured FRF. These are designated in the figure as ‘contaminants’. Contaminants were identified as frequency peaks that remained independent of the partial immersion of the hydrofoil. The primary contaminant is thought to be resonance of the bracket used to attach the shaker motor to the hydrofoil, which occurs between 50 Hz and 55 Hz. Over the course of testing, gradual loosening of threaded braces on the bracket caused a slow decreasing trend in its resonant frequencies. A second significant source of contamination occurred with vibration of the transverse rails (shown in figure 3), to which the hydrofoil set-up was attached. The rails, which acted as simply supported beams, began vibrating in a fundamental vertical mode at 38 Hz (confirmed by accelerometer measurements of the rails). Some additional contamination of the spectrum near $f=100~\text{Hz}$ is due to another resonant mode of the cavitation channel structure, and is compounded by the second harmonic of the AC line noise at 50 Hz. The thick grey line plotted over the measured FRF represents principal-component reconstruction of the compliance FRF, yielded by applying summing equation (4.1) over the four resonances of interest (omitting the contaminants). Finally, it should be noted that the peaks of the individual modes in the FRF are indicative of the damped natural frequencies of the respective modes, which are slightly smaller than the undamped experimental frequencies listed in table 2.

Figure 6. FRF decomposition for the hydrofoil in dry conditions. The vertical grey lines indicate modal frequencies predicted by the FEA model. Data collected for 1660 s at 500 Hz. Contaminant modes include vibration of the shaker motor mount at approximately 50 Hz, facility modes at 38 Hz and approximately 100 Hz and AC line noise and harmonics thereof at multiples of 50 Hz.

An FEA modal analysis was performed in ANSYS Mechanical^®. The details of the model are contained in appendix A. The comparison between the modes predicted by the FEA model and those estimated by EMA are shown in tabular form in table 2 and the resulting mode shapes are compared in figure 7. The agreement is generally good, although FEA tends to over-predict the frequencies of higher modes. The FEA model accounted for the compliance of the load cell and the root clamp, but there remains some small compliance in the vibration test structure to which the load cell is mounted that departs from the simulated assumption of a perfect clamp at the base of the load cell. Additionally, some frictional losses are present in bolted connections that are not captured by the undamped modal analysis. Both of these observations contribute to the differences between FEA and EMA. Nevertheless, modal frequencies are in close agreement, with a maximum error of 6.8 %. The associated mode shapes are dimensionless, so apparent differences between the magnitude of simulated and experimental modes are of no physical significance. The number and orientation of nodal lines are in good agreement, with the exception of the third ordered mode (X-Bend 2). The nodal line for the third mode lies at a different angle than that of the predicted mode shape. The contaminant mode attributed to the shaker motor mounting bracket is a dominant component in the FRF between modes 2 and 3, while mode 3 appears significantly less energetic. The proximity of the contaminant mode is therefore hypothesized to cause a flawed mode shape estimation. Fortunately, the global frequency and damping ratio estimates are less affected by the nearby contaminant than is the residue matrix, so the other modal parameters are thought to remain valid.

Table 2. Comparison of undamped natural frequencies ( $f_{0}$) and mode shape descriptions between FEA and EMA. Note that mode 4 (lead lag) is not observed experimentally because the shape-sensing spars are designed to measure lateral bending strains only.

Figure 7. Comparison of in vacuo mode shapes predicted by FEA model and extracted by EMA. The experimental mode shapes have been placed in the same orientation as the FEA figures. Qualitative agreements are good, though the nodal line of the X-Bend 2 mode does not match between the experiment and the simulation. Experimental modes 1, 2, 3 and 5 were measured in the CNR INM channel in dry conditions, using a frequency sweep with a step size of 0.05 Hz and $N_{C}=15$ cycles at each discrete frequency. Mode 6 was measured on the vibration frame at the University of Michigan, using a continuous linear frequency sweep.

4.2.2 Effects of partial immersion in still water

In figure 8, the resonant frequencies of all modes at $AR_{h}=1.0$ in still water decrease compared to the dry condition. This reduction occurs as a result of increased fluid added mass, which causes inertial resistance to the motion of the hydrofoil. The contaminating modes, however, remain fixed (one of the ways in which the contaminants were identified and separated from the modes of interest). Those contaminant modes are better separated from the modes of interest in quiescent water, particularly near the third (X-Bend 2 $+$ Twist) mode. No differences are immediately obvious in the widths of the individual peaks, compared to figure 6, which suggests that the damping ratio ($\unicode[STIX]{x1D709}_{e}$) is not strongly affected by partial immersion in quiescent liquid.

Figure 8. FRF decomposition for the hydrofoil in still water $\unicode[STIX]{x1D6FC}=10^{\circ }$; $Fn_{h}=0$; $AR_{h}=1.0$. Data collected for 4500 s at 500 Hz.

4.2.3 Effects of forward speed and flow regime

Figure 9 depicts the FRFs for the hydrofoil in typical FW, FV and PC flows alongside the FRF for a quiescent fluid. The top plot contains the experimental FRF, while the bottom plot contains the FRF curve fits, reconstructed from the identified modes; individual fitted modes are omitted for clarity. The spectra for each of the flow regimes with individual modal contributions are shown in figures 25–27 in appendix B.

Figure 9. FRFs of the flexible hydrofoil in quiescent water, FW flow, FV flow and PC flow. The top plot depicts the (smoothed) experimental FRFs. The lower plot depicts the de-noised FRFs synthesized from the fitted modes, excluding contaminant modes. Inset photographs show the flow pattern in each flow regime. Note, in particular, the photographs from two instants in PC flow that demonstrate the unsteadiness in the leading edge cavity. All data were collected at a sampling rate of 500 Hz. Recording durations were as follows. FW flow: 520 s. FV flow: 1900 s. PC flow: 930 s.

In FW flow with forward speed, additional flow-induced motion and facility vibrations are present in the collected time series, resulting in a noisier spectrum. Modal peaks are still successfully extracted, and comparison with the quiescent immersion spectrum in figure 9 shows a reduction in the magnitude and some visible broadening of the individual peaks, which suggests that some additional damping may be introduced by the hydrodynamic contribution of vortex shedding, wave generation and spray. Interestingly, modal frequencies appear lower in FW flow than in quiescent conditions, indicating some effect of the flow upon the hydrodynamic stiffness, hydrodynamic damping or both. Additional low-frequency unsteadiness in the cavitation channel, created by sloshing, eddying inlet flow and a small periodic velocity surge, cause increased spectral content at frequencies from 0–5 Hz, reducing the coherence of the hydrofoil’s response in the range of mode 1. As a result, the first mode became increasingly difficult to fit at higher speeds.

In FV flow at the same $Fn_{h}$ and $AR_{h}$, the spectrum appears noisier than in the FW case, though this is due in part to a longer sampling duration and resulting finer frequency resolution. The contaminant modes are no longer very clear, and no attempt was made to identify them. The modal peaks are shifted to higher frequencies than in the FW case, signifying a probable reduction in added mass. In most cases, however, the FV damped modal frequencies remain smaller than those in quiescent conditions, again signifying appreciable changes in the fluid modal forces.

The noise level for the PC regime is similar to that in the FV flow regime, but the peaks are less distinct. The vaporous cavity at the leading edge was found to be highly unsteady. This unsteady partial shedding of the vaporous cavity has been shown to induce temporal modulations of the fluid mass, damping and stiffness matrices (Akcabay et al. Reference Akcabay, Chae, Young, Ducoin and Astolfi2014; Akcabay & Young Reference Akcabay and Young2015; Lelong, Guiffant & Astolfi Reference Lelong, Guiffant and Astolfi2017), resulting in non-stationary vibrations. The spectrum in figure 9 therefore represents an average of the time-varying spectra. While leading edge vaporous partial cavities were unsteady, they lacked coherent shedding patterns or well-defined shedding periods, probably because the cavities were quite short ($L_{C}\lesssim 0.4C$ at their longest points, as shown by the inset of figure 9), and because significant spanwise variation in the cavity length caused predominantly incoherent 3-D cavity shedding. For more discussion of the cavity dynamics, see Part 1.

It should be noted that, unlike the flow-excited spectra presented in Part 1 of this paper, no evidence of the von Kármán wake excitation is found in the FW spectrum. The dynamic wake excitation is still present, but the greatly increased input power of the shaker motor (and the associated forced response) make the effects of the ambient excitation negligibly small when the ratio of output spectrum to input spectrum is taken (see (4.3)). This was confirmed by a comparison between the spectral powers of the hydrodynamic loads on the hydrofoil and the spectral power of the shaker excitation force. In the frequency band associated with von Kármán shedding, the spectral power of the ambient (flow-induced) excitation was found to be two to three orders of magnitude smaller than that of the shaker motor when excitation was enabled.

4.4.1 Effects of varying immersion depth in quiescent water

Figure 11 shows the modal frequencies for modes 1, 2, 3 and 5 as functions of $AR_{h}$. Data may also be found in table 5 in appendix C. Recall that mode 4, alternatively known as Y-Bend 1 or the lead-lag mode, was not measured. Data from the ground-fixed frame at the University of Michigan are plotted in the range of $0\leqslant AR_{h}\leqslant 2.2$ as filled symbols. Data from CNR INM are plotted for $AR_{h}=0$ and $AR_{h}=1.0$ as open symbols. Comparison of the two datasets reveals a mean difference of 0.53 Hz and a maximum percentage error of 2.9 %, indicating an acceptably small facility bias.

The resonant frequencies of all modes in partially immersed conditions (quiescent water, FW, FV and PC flows) are substantially lower than those in the dry conditions. Dense liquid presents an inertial resistance to the structural accelerations, expressed in (2.4) as the fluid added-mass matrix $\unicode[STIX]{x1D648}_{\unicode[STIX]{x1D65B}}$. Additionally, fluid inertial forces most strongly oppose motion normal to faces with large projected areas. Thus, the first bending mode is affected most substantially by immersion in water. The same conclusion was reached by Harwood et al. (Reference Harwood, Stankovich, Young and Ceccio2016a), Phillips et al. (Reference Phillips, Cairns, Davis, Norman, Brandner, Pearce and Young2017). Modal coalescence between modes 2 and 3 appears to occur at an immersed ratio of $AR_{h}=2.2$. Coalescence in a quiescent fluid, unlike that in a flowing one, will not yield the coupled flutter instability described in § 2 because no hydrodynamic damping is present at zero speed to facilitate the transfer of energy between coincident modes. Note that single-reference testing (one excitation point) makes it difficult to discern individual mode shapes near the coalescence condition.

4.4.2 Effects of varying flow speed and flow regime

Undamped modal frequencies are shown in figure 12 as functions of the flow regime and $Fn_{h}$ for modes 1, 2, 3 and 5. Plots to the right contain the same data, but with each mode plotted on magnified axes for clarity. Filled symbols indicate the ensemble average of all replications and all individual degree-of-freedom FRFs at each condition. Vertical bars indicate $\pm$ one standard deviation of the aggregate samples, with statistical outliers removed using an iterative Grubbs test at 5 % significance (Grubbs Reference Grubbs1969). Data may also be found in table 6 in appendix C.

Figure 12 demonstrates that modal frequencies are higher in FV flow than in FW or PC flow for all modes. In FV flow, a substantial portion of the dense fluid (water) is displaced by air (the precise proportion is a function of the cavity’s size, and thus depends upon all the attending physics). As a result, the fluid added mass, which is proportional to the fluid density, is significantly reduced. The frequencies in the PC flow are slightly higher than the FW data points, but remain within one standard deviation. Despite the fact that water vapour is lighter than liquid water by five orders of magnitude, the maximum length of the cavity at the mid-span was less than 40 % of the chord length in all PC flows observed, and the vapour fraction within the cavitating region was likely less than $25\,\%$. Thus, the density field was modified only over a small portion of the hydrofoil suction surface, and by a smaller density variation than in the case of FV flow, so it is unsurprising that the effects upon resonances are less pronounced in PC flow than in FV flow. The small sizes of the vaporous cavities are a consequence of the proximity of the free surface. Larger cavities tended to rupture the free surface seal through the development of Taylor instabilities (Taylor Reference Taylor1950; Rothblum, Mayer & Wilburn Reference Rothblum, Mayer and Wilburn1969; Swales et al. Reference Swales, Wright, McGregor and Rothblum1974; Young et al. Reference Young, Harwood, Miguel, Francisco, Jacob and Ceccio2017) and transition to FV flow. We would expect to see increasing effects of vaporous cavitation if the sizes of the cavities were greater. At the limit of a vaporous super-cavity, we would expect to see results that closely match those of FV flow.

There is a small decreasing trend in the FW resonant frequencies with increasing Froude number, but the trend is non-monotonic. Results from Besch & Liu (Reference Besch and Liu1973) demonstrated a similar behaviour, with a weak initial decrease in modal frequencies, followed by an increase at higher speeds. This behaviour suggests that the fluid exerts a disturbing, rather than restoring, force in phase with the structural displacements – thus leading to the notion of a hydrodynamic stiffness that is conjectured to be weakly negative. This corroborates the results of Part 1 of this paper. On the other hand, frequencies corresponding to FV flow increase measurably with increasing speed in figure 12. The length of a ventilated cavity increases with the forward speed of the hydrofoil, indicating that as $Fn_{h}$ increases, the growing cavity displaces water over an increasingly large area of the hydrofoil’s suction surface. The decrease in local mean fluid densities leads to an increase in the modal frequencies and a reduction in the fluid inertial loading. These concepts will be explored further in § 5.3.3.

4.5.1 Damping mechanisms in the fluid–structure system

Damping remains one of the most challenging aspects of vibrations in any context, due to the many forms and mechanisms of energy dissipation. Blake (Reference Blake1972), Blake & Maga (Reference Blake and Maga1975) explored, through theory and experiments, the roles of acoustic, material, mechanical and viscous dissipation paths. In a heterogeneous system, such as a structure vibrating in multiphase flow, the complexity increases further with the addition of fluid interfaces, latent thermal energies associated with phase change in cavitating flows or the entrainment of gas from the free surface.

The summed damping matrix $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D65A}\unicode[STIX]{x1D65B}\unicode[STIX]{x1D65B}}=(\unicode[STIX]{x1D63E}+\unicode[STIX]{x1D642}_{\unicode[STIX]{x1D668}}\unicode[STIX]{x1D64E}^{D})$ (see (2.6)) is assumed to be a summation of linearized damping components, each representing such a distinct physical mechanism,

(4.14)$$\begin{eqnarray}\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D65A}\unicode[STIX]{x1D65B}\unicode[STIX]{x1D65B}}=\mathop{\sum }_{i}\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D65E}}.\end{eqnarray}$$

The following damping terms ($\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D65E}}$) are expected to occur in the multiphase flows investigated in this work:

1. (a) Material dissipation $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D662}\unicode[STIX]{x1D659}}$: viscoelastic and frictional forces in solid materials dissipate some percentage of strain energy to thermal energy during each cycle of vibration. This component was represented by $\unicode[STIX]{x1D642}_{\unicode[STIX]{x1D668}}$ in § 2.

2. (b) Mechanical frictional damping $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D65B}}$: minute frictional forces in bolted joints or at other contact interfaces in non-monolithic mechanical structures dissipate energy (Coulomb damping).

3. (c) Facility dissipation $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D65B}\unicode[STIX]{x1D659}}$: the test specimen and the facility are coupled via the interceding fluid and mechanical coupling, such that some power will be transmitted to the facility and dissipated to distributed material and mechanical losses.

4. (d) Viscous and eddy damping $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D66B}}$: in a fluid at any speed, the motion of a structure will create eddies or shear layers that dissipate energy via fluid shear stresses. Such viscous damping is poorly understood (Blake & Maga Reference Blake and Maga1975), but some insight into the functional form of the viscous damping coefficient can be gleaned from semi-empirical expressions for viscous drag. For example, an expression for the viscous damping force on a degree of freedom $i$ with an associated elemental area $dA_{i}$ may be derived from the second term of the Morison equation (Morison, Johnson & Schaaf Reference Morison, Johnson and Schaaf1950; Newman Reference Newman2017)

   (4.15)$$\begin{eqnarray}f_{v,i}=-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}_{f}C_{D}dA_{i}v_{i}\sqrt{U^{2}+v_{i}^{2}},\end{eqnarray}$$
   where $v_{i}$ is the velocity of degree of freedom $i$ and $C_{D}$ is an empirical drag coefficient. The subscripted $i$ on both velocity and force indicates that the associated viscous damping coefficient would be located along the diagonal of $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D66B}}$. Under the assumption of harmonic motion with nodal amplitude $x_{0,i}$, this damping element becomes,
   (4.16)$$\begin{eqnarray}C_{v,ii}={\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}C_{D}dA_{i}\sqrt{U^{2}+\unicode[STIX]{x1D714}^{2}x_{0,i}^{2}}.\end{eqnarray}$$
   The damping coefficient will thus increase in magnitude with forward speed, asymptotically approaching a linear trend at higher speeds. For small speeds, a second-order Taylor series of (4.16) about $U=0$ yields,
   (4.17)$$\begin{eqnarray}C_{v,ii}=\frac{1}{2}\unicode[STIX]{x1D70C}C_{D}dA_{i}\left(\unicode[STIX]{x1D714}x_{0,i}+\frac{U^{2}}{2\unicode[STIX]{x1D714}x_{0,i}}\right).\end{eqnarray}$$
   At zero speed, the damping coefficient increases linearly with the amplitude of motion – a trend observed experimentally by Phillips et al. (Reference Phillips, Cairns, Davis, Norman, Brandner, Pearce and Young2017). For small values of $U$, the damping should increase quadratically as $U$ increases.

5. (e) Wake damping $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D66C}}$: circulatory forces in opposition to structural velocities, caused by the shedding of vorticity into hydrofoil wake at non-zero speed. This inviscid source of damping is often modelled using classic theoretical solutions such as the Theodorsen or Wagner function (Wagner Reference Wagner1925; Theodorsen Reference Theodorsen1935). The damping force, non-dimensionalized as the loss ratio, was shown both theoretically and experimentally to increase linearly with forward speed, and inversely with the reduced frequency by Blake & Maga (Reference Blake and Maga1975).

6. (f) Radiation damping $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D667}}$: inviscid radiation of acoustic and free-surface waves occurs at all speeds.

   1. (f.1) Acoustic radiation damping is a product of the acoustic environment; some of the energy deposited in acoustic waves are dissipated within the fluid, while the rest is transmitted to the facility to be dissipated by facility damping. The effect of flow speed on acoustic damping is expected to be negligible, as the acoustic speed is orders of magnitude higher than the maximum mean flow velocity achieved during testing. The exception may be where bubbly flows decrease the bulk modulus of the mean fluid, which can cause the acoustic speed of the fluid mixture to approach the flow speed.

   2. (f.2) Free-surface waves will be radiated away from the body whenever vibrations occur at or near the free surface. By linear theory, the wave height and wave patterns will depend upon the mode shape and the wavelength will be a function of the vibration frequency, such that both affect the rate of energy dissipation – and hence damping. Radiation of surface waves may also occur on the cavity walls in multiphase flows. Linear potential theory predicts that wave radiation damping is zero for body motions at zero and infinite frequencies, with a maximum at an intermediate frequency that depends upon the geometry of the body. Forward speed modifies the potential flow boundary value problem, so wave radiation damping is speed dependent, as shown in Chakrabarti (Reference Chakrabarti2002) and Faltinsen (Reference Faltinsen2005).

7. (g) Cavity damping $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D658}}$: dissipative paths in multiphase flow include the bubbly breakup and shedding of cavities (either vaporous or ventilated) and the vaporization and condensation of localized regions in vaporous cavitating flow. Cavity damping also includes dissipation in the substantial spray sheet created by ventilated cavities.

Linearization and summation of the terms above leads to the following expression:

(4.18)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D63E} & = & \displaystyle \unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D662}\unicode[STIX]{x1D659}}+\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D65B}}+\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D65B}\unicode[STIX]{x1D659}}+\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D66B}}+\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D66C}}+\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D667}}+\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D658}}\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D64E}}+\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D64C}}+\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D643}},\end{eqnarray}$$

where $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D64E}}$, $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D64C}}$ and $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D643}}$ are respectively dubbed the structural, quiescent and hydrodynamic damping matrices. The latter expression is proposed as an alternate, simplified grouping of the former terms.

1. (i) Structural damping,

   (4.19)$$\begin{eqnarray}\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D64E}}=\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D662}\unicode[STIX]{x1D659}}+\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D65B}},\end{eqnarray}$$
   includes all material and frictional losses to the hydrofoil, mounting hardware and test environment when the hydrofoil is vibrating in a dry configuration. This damping force is assumed to be independent of the flow regime.

2. (ii) Quiescent fluid damping accounts for structural losses via acoustic transmission to the test environment, free-surface wave radiation and viscous eddy generation,

   (4.20)$$\begin{eqnarray}\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D64C}}=\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D65B}\unicode[STIX]{x1D659}}+\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D66B}_{\unicode[STIX]{x1D664}}}+\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D667}_{\unicode[STIX]{x1D7EC}}},\end{eqnarray}$$
   where $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D667}_{\unicode[STIX]{x1D7EC}}}$ is the component of the radiation damping ($\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D667}}$) present at zero speed; $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D66B}_{\unicode[STIX]{x1D7EC}}}$ is the speed-independent component in the Taylor series expansion of $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D66B}}$, which varies with frequency. By the arguments above, the speed-independent component of the viscous damping should also be of the form,
   (4.21)$$\begin{eqnarray}\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D66B}_{\unicode[STIX]{x1D7EC}}}\propto \unicode[STIX]{x1D714}\boldsymbol{X}.\end{eqnarray}$$

3. (iii) Hydrodynamic damping includes the increment (or decrement) to wave radiation and viscous damping with forward speed. It also includes inviscid wake damping for all flow regimes and cavity damping for multiphase flow regimes. The wake damping $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D66C}}$ is a function of reduced frequency, and for a constant reduced frequency, is known to vary linearly with speed, i.e. $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D66C}}\propto U$. The viscous damping $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D66B}}$ varies quadratically with speed, i.e. $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D66B}}\propto U^{2}$. $\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D667}}$ varies with the speed in a geometry- and frequency-dependent manner. Combining all the terms, the hydrodynamic damping becomes,

   (4.22)$$\begin{eqnarray}\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D643}}=\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D66C}}+\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D66B}}+\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D667}}+\unicode[STIX]{x1D63E}_{\unicode[STIX]{x1D658}}.\end{eqnarray}$$

Table 3 summarizes the various damping mechanisms within each damping group. Columns on the right denote the flow regimes, and shaded cells indicate which mechanisms are present in each flow regime. Hatching indicates that the damping mechanism is expected to vary as a function of the speed within that particular flow regime. For example, FW flow is seen to dissipate energy via dry structural damping (equal to that in dry conditions), quiescent fluid damping and hydrodynamic damping. The latter-most includes the changes in wave radiation and viscous damping wrought by the non-zero flow velocity, as well as the addition of the inviscid wake damping. All mechanisms in this third group are expected to vary as functions of the flow velocity and frequency.

Table 3. Breakdown of specific damping mechanisms into structural, quiescent and hydrodynamic damping groups for each flow regime: dry conditions, quiescent immersion and FW, FV and PC flows. Coloured shading indicates which physical mechanisms are encompassed by a group in the respective flow regime. Hatching indicates that the respective mechanism is expected to be dependent upon speed within that flow regime.

Figure 13. Variation in the effective critical damping ratios with immersed aspect ratio. The data collected in the cavitation channel agree closely with those collected with the free-standing frame, indicating a lack of significant influence of the test environment upon the modal parameters. A general increase in critical damping ratio is observed with increasing immersion of the hydrofoil, presumably due to an increase in the role that viscous dissipation plays as a greater proportion of the hydrofoil is immersed, combined with the amplitude and frequency dependence of the viscous damping.

Figure 14. Equivalent total damping ratios of the hydrofoil as functions of Froude number in FW flow ($0^{\circ }\leqslant \unicode[STIX]{x1D6FC}\leqslant 12^{\circ }$), FV flow ($7^{\circ }\leqslant \unicode[STIX]{x1D6FC}\leqslant 12^{\circ }$) and PC flow ($\unicode[STIX]{x1D6FC}=7^{\circ },12^{\circ }$ and $0.42\leqslant \unicode[STIX]{x1D70E}_{v}\leqslant 0.85$). All data (except dry conditions) are for $AR_{h}=1.0$ and all measurements were made in test environment 1 – the CNR INM cavitation channel.

In summary, damping occurs through material hysteresis, mechanical connections in the model and facility vibrations (collectively referred to as structural damping); acoustic radiation, wave radiation and viscous effects (collectively called quiescent fluid damping); and viscous shear layer generation, free-surface wave generation, vortex shedding and cavity damping (hydrodynamic damping). The equivalent damping ratio, $\unicode[STIX]{x1D709}_{e}$, is representative of the aggregate damping of all energy dissipation mechanisms, represented by a single linear dashpot.

4.5.2 Effects of varying immersion depth in quiescent water

Effective damping ratios are plotted in figure 13 as functions of the immersed aspect ratio for each mode. Once again, data for the quiescent fluid were collected in both test environments. Data from the University of Michigan are plotted in the range of $0\leqslant AR_{h}\leqslant 2.2$. Data from CNR INM at $AR_{h}=0$ and $AR_{h}=1.0$ (plotted as open symbols) again agree well with the measurements from Michigan. The same data are given in table 7 in appendix C. Modal damping in dry conditions (where $\unicode[STIX]{x1D709}_{e}\approx \unicode[STIX]{x1D709}_{s}$) is between 1.2 % and 1.7 % for each mode. The near constancy of the damping across all modes suggests frequency independence – a feature of structural hysteretic damping.

There is a general increasing trend in $\unicode[STIX]{x1D709}_{e}$ with increasing values of $AR_{h}$ for all modes, although a majority of the data reside between 1 % and 2 % critical damping. In fact, the damping ratios for modes 2, 3 and 5 are virtually unaffected by the initial immersion to $AR_{h}=0.5$, while the damping ratio of mode 1 increases substantially with initial immersion. Further increasing the immersion depth yields non-monotonic increases in the damping across all modes, which we attribute to several causes.

Acoustic radiation damping is expected to be small and wave radiation damping is driven by the amplitude of motion near the free surface (local amplitudes generally decrease as one moves from the tip toward the root, as shown in figure 7). The changes are thus probably caused by nonlinear viscous damping in the fluid. As immersion increases, a greater portion of the hydrofoil is subjected to the larger viscous damping force. At the same time, the entrained mass of the fluid with the vibrating structure is increasing, and with it increases the total energy contained in a cycle of vibration. The critical damping ratio is proportional to the percentage of energy dissipated per cycle, so even while the viscous dissipation rate increases, the proportional dissipation rate does not necessarily follow the same trend. Additionally, equation (4.2) suggests that as the natural frequency is reduced for a mode, the magnitude of the FRF increases, which leads to a second conjecture. Shaker motor excitation forces were maintained at approximately the same amplitude at each immersed depth, and greater resulting motion amplitude might be expected to yield increasing viscous damping – as suggested by (4.17) and shown experimentally by Phillips et al. (Reference Phillips, Cairns, Davis, Norman, Brandner, Pearce and Young2017). However, those increased amplitudes are – at the same time – partially offset in (4.17) by the diminished frequency, which should reduce the nonlinearity of the viscous damping. The sudden increase in damping near $AR_{h}=1.8$ is not fully explained by these behaviours, however, and further systematic tests will be required in the future to explore it. At $AR_{h}=2.2$, at which point the frequencies of modes 2 and 3 coalesce, the damping of mode 2 increases while that of mode 3 decreases, which suggests an exchange of energy from the former to the latter.

4.5.3 Effects of varying flow speed and flow regime

The equivalent viscous damping ratio ($\unicode[STIX]{x1D709}_{e}$) is plotted for all conditions at an immersed aspect ratio of $AR_{h}=1.0$ in figure 14. The same data are given in table 8 in appendix C. The estimates of in situ damping exhibit significantly more scatter than those of modal frequencies. Damping values are extremely sensitive to small changes in the FRF, and are notoriously difficult to measure consistently (Soroka Reference Soroka1949; Bert Reference Bert1973; Reese Reference Reese2010).

In FW flow, modes 2, 3 and 5 exhibit increasing damping with forward speed – only for mode 2 is the increase monotonic. The damping of mode 1 initially increases to a maximum at $Fn_{h}=1$ before decreasing until a minimum is reached at $Fn_{h}=2.5$. The sharp decrease in damping for mode 1 at higher speeds is suspected to be a spurious result of flow-induced vibrations. It was mentioned in Part 1 that eddying flow in the channel caused unsteady inflow conditions at frequencies of $f\lesssim 5~\text{Hz}$. As a result, there is spectral energy in the neighbourhood of mode 1 that degrades the coherence and can be erroneously inferred by the modal analysis procedure as vanishingly small damping.

FV and PC flows are conjectured to affect the damping in several ways. First, the reduction in the viscosity of a portion of the surrounding flow is expected to suppress a portion of the viscous damping, though the degree of suppression will depend upon both the mode shape and the extent of the cavity. Second, the cavity boundary presents another gas–liquid interface on which radiated waves may be generated. Third, the presence of the cavity will alter the interplay between the hydrofoil wake and the pressure distribution on the hydrofoil by attenuating and delaying the transmission of pressure from shed vortices. Damping in the FV and PC flow regimes is approximately equal to that in the FW flow regime for mode 1, although the preceding discussion suggests that damping estimates for mode 1 are suspect. In modes 2, 3 and 5, the damping in the FV flow regime increases with increasing mode number, relative to the FW damping. Within each mode, the FV data show only a weakly negative trend that contrasts with the increasing trend in the FW data. This may be explained by the overall growth in the size of the cavity as the speed increases, which will reduce the hydrodynamic damping due to the wake. At the same time, modes of increasing order will generate shorter radiated wavelengths, such that a greater rate of dissipation to the cavity walls is conceivable. In PC flow, localized evaporation and condensation may be expected to contribute to the cavity damping. The lack of a consistent trend in PC damping may be attributed to both variability in the vaporous cavitation number $\unicode[STIX]{x1D70E}_{v}$ not captured by the plots of figure 14 and to the time modulation of the system parameters by the unsteady cavity (described in § 2).

Despite the observed trends with changing immersion, speed and flow regimes, conclusions cannot yet be drawn regarding the damping groups outlined in § 4.5 because, as will be shown in the following section, a quantitative comparison cannot be made between the damping groups of table 3.

5.2.1 Modelling assumptions

In FW flow, the generalized stiffness ratio may be uniquely determined because the added-mass ratio is known from testing in a quiescent fluid. In multiphase flow, on the other hand, it was posited in the preceding section that the stiffness ratio should be modelled as,

(5.8)$$\begin{eqnarray}\frac{K_{f,n}^{m}}{K_{s,n}^{m}}=C\left.\frac{K_{f,n}^{m}}{K_{s,n}^{m}}\right|_{FW}.\end{eqnarray}$$

Part 1 of this paper series showed that for a simplified 2D-2DOF model, the hydrodynamic stiffness was closely linked to the slope of the lift coefficient for bending motions and the slope of the moment coefficient for twisting motion – defined in both cases with respect to changing angle of attack. Moreover, it was shown that the lift and moment coefficients, normalized by the values in the FW flow regime, can be correlated with the cavitation parameter $\unicode[STIX]{x1D70E}_{c}/\unicode[STIX]{x1D6FC}$, where $\unicode[STIX]{x1D70E}_{c}$ is the general cavitation number. Defining $\unicode[STIX]{x1D70E}_{c}$ at the middle of the immersed span ($z^{\prime }=h/2$) and fixing the cavity pressure as $P_{C}=P_{0}$ for FV flows and $P_{C}=P_{v}$ for FW and PC flow, the cavitation parameter may be written,

(5.9a)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x1D70E}_{c}}{\unicode[STIX]{x1D6FC}}=\frac{\unicode[STIX]{x1D70E}_{v}}{\unicode[STIX]{x1D6FC}}+\frac{1}{\unicode[STIX]{x1D6FC}{Fn_{h}}^{2}}\quad \text{FW and PC flows}, & \displaystyle\end{eqnarray}$$

(5.9b)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x1D70E}_{c}}{\unicode[STIX]{x1D6FC}}=\frac{1}{\unicode[STIX]{x1D6FC}{Fn_{h}}^{2}}\quad \text{FV flows}. & \displaystyle\end{eqnarray}$$

It is proposed that the same correlation equations proposed in Part 1 can thus be used for $C$ in (5.8). For bending-dominated modes (modes 1 and 5), the equation for $C_{L}$ is then,

(5.10)$$\begin{eqnarray}\displaystyle C_{1,5}=\frac{K_{f,n}^{m}}{K_{f,n}^{m}|_{FW}} & {\approx} & \displaystyle \frac{C_{L}}{C_{L,FW}}\nonumber\\ \displaystyle & = & \displaystyle \frac{\displaystyle \left(\frac{\unicode[STIX]{x1D70E}_{c}}{\unicode[STIX]{x1D6FC}}\right)^{2}-0.935\left(\frac{\unicode[STIX]{x1D70E}_{c}}{\unicode[STIX]{x1D6FC}}\right)+1.1}{\displaystyle \left(\frac{\unicode[STIX]{x1D70E}_{c}}{\unicode[STIX]{x1D6FC}}\right)^{2}-1.535\left(\frac{\unicode[STIX]{x1D70E}_{c}}{\unicode[STIX]{x1D6FC}}\right)+2.03}.\end{eqnarray}$$

The same approach, applied to data for the yaw moment coefficient ($C_{M}$) yields the stiffness ratio for twisting-dominated modes (modes 2 and 3),

(5.11)$$\begin{eqnarray}\displaystyle C_{2,3}=\frac{K_{f,n}^{m}}{K_{f,n}^{m}|_{FW}} & {\approx} & \displaystyle \frac{C_{M}}{C_{M,FW}}\nonumber\\ \displaystyle & = & \displaystyle \frac{\displaystyle \left(\frac{\unicode[STIX]{x1D70E}_{c}}{\unicode[STIX]{x1D6FC}}\right)^{2}+0.0328\left(\frac{\unicode[STIX]{x1D70E}_{c}}{\unicode[STIX]{x1D6FC}}\right)+1.0721}{\displaystyle \left(\frac{\unicode[STIX]{x1D70E}_{c}}{\unicode[STIX]{x1D6FC}}\right)^{2}+0.767\left(\frac{\unicode[STIX]{x1D70E}_{c}}{\unicode[STIX]{x1D6FC}}\right)+3.659}.\end{eqnarray}$$

The resulting values of $C$ are shown in figure 15. Note that the data shown are simply values computed from (5.10) and (5.11) and plotted according to the respective flow regime.

Figure 15. Values of $C$ for modes 1, 2, 3 and 5 for use in (5.8). Note that the data shown are the result of an empirical model, and are not extracted directly from experimental measurements.

5.2.2 Effects of varying flow speed and flow regime

The ratios of generalized hydrodynamic stiffness to generalized structural stiffness are shown in figure 16. Only the results in the FW regime are computed directly from measured modal parameters. The FW fluid stiffness generally becomes increasingly negative with increasing speed, especially for mode 1. This effect is attributed to the location of the lift centre upstream of the elastic axis on the hydrofoil. This eccentricity reduces the effective stiffness of the system by amplifying any initial displacement that increases the effective angle of attack of the hydrofoil, as was described in Part 1. Present in all modes, but more pronounced in modes 3 and 5, is a reversal in the trend of the stiffness ratio. For modes 2, 3 and 5, however, the stiffness ratio never exceeds 6 %, so one should not assign too much significance to these small variations. The hydrodynamic stiffness in the FV and PC flow regimes follow directly from (5.8), so no conclusions should be drawn from them alone. However, the data in the FV regime, in particular, support the results from Part 1. Namely, the magnitudes of the FV hydrodynamic stiffness values are smaller than those in the FW regime, and the difference increases as the Froude number increases.

Mode 1, while primarily a bending mode, does involve a small amount of positive twist, which elicits the hydroelastic coupling described. In the cases tested, the fluid stiffness never exceeds 25 % of the structural value, indicating no immediate risk of static divergence. If the speed were further increased, the generalized structural stiffness of mode 1 would eventually be negated, causing an unbounded growth in the modal participation factor, consistent with the bending-induced failure of most lifting surfaces experiencing static divergence. Note that this behaviour is usually traced to the off-diagonal bend–twist coupling terms in $\unicode[STIX]{x1D646}_{\unicode[STIX]{x1D65B}}$, as in Part 1 or Chae et al. (Reference Chae, Akcabay and Young2017), Young et al. (Reference Young, Garg, Brandner, Pearce, Butler, Clarke and Phillips2018a). In this case, the modal decoupling instead relates the stiffness to the participation of individual modes. Thus, higher-order modes will have increasingly complex loading behaviours, especially if nonlinear and 3-D lifting-surface effects are considered. The complex mode shapes may also be to blame for the non-monotonic behaviours in figure 16.

Figure 16. Ratio of generalized fluid (hydrodynamic) stiffness to generalized structural stiffness as function of $Fn_{h}$ in FW flow ($0^{\circ }\leqslant \unicode[STIX]{x1D6FC}\leqslant 12^{\circ }$), FV flow ($7^{\circ }\leqslant \unicode[STIX]{x1D6FC}\leqslant 12^{\circ }$) and PC flow ($\unicode[STIX]{x1D6FC}=7^{\circ },12^{\circ }$ and $0.42\leqslant \unicode[STIX]{x1D70E}_{v}\leqslant 0.85$). All data (except dry conditions) are for $AR_{h}=1.0$. Only the values in the FW regime are calculated directly; those in the FV and PC regimes are modelled using (5.8), (5.10) and (5.11). The magnitude of the fluid stiffness in the FV regime is smaller than that in FW flow, and the proportional difference increases with increasing $Fn_{h}$.

The correlation between the generalized fluid stiffness ratio and the cavitation parameter was found to be poor in the FW flow regime, and is not shown here. This occurs because the angle of attack resides in the denominator of $\unicode[STIX]{x1D70E}_{c}/\unicode[STIX]{x1D6FC}$, to which the fluid stiffness appears relatively insensitive in FW flow.

5.3.1 Effects of varying immersion depth in quiescent water

Figure 17 shows the changes the in generalized added mass ratios with increasing values of $AR_{h}$ in a quiescent fluid. Data from both the CNR INM cavitation channel and the free-standing test frame at the University of Michigan are represented, with excellent agreement between the two facilities. The added-mass ratio monotonically increases across all modes as the hydrofoil is immersed in water. As posited before, this behaviour is due to the inertial loading of the dense fluid over an increasing proportion of the structure. Additionally, the added-mass ratio is largest for mode 1 and decreases with ascending mode number (for $AR_{h}<1.8$). This observation occurs because the fluid inertia most strongly affects those modes whose mode shapes describe large swept volumes. For $AR_{h}\geqslant 1.8$, the added-mass ratio for mode 2 (Z-Twist 1) surpasses that for mode 3 (X-Bend 2). Considering the associated mode shapes in figure 7, it stands to reason that the near alignment between the nodal line and $Z$-axis for mode 2 leads to a linear increase in the fluid added-mass coefficient with increasing depth. In contrast, mode 3 possesses a nodal line more nearly parallel to the water surface; as $AR_{h}$ approaches unity, the node line becomes immersed, at which point incremental changes in $AR_{h}$ do not strongly modify the entrained mass. As the depth continues to increase, and the node line becomes submerged, the newly submerged portions of the structures undergo increasing amplitudes of motion, entraining more fluid mass, which leads to the positive curvature in the added-mass trend for mode 3.

Figure 17. Ratios of generalized fluid mass to generalized structural mass for modes 1, 2, 3 and 5 as functions of $AR_{h}$. Closed symbols denote data collected in the University of Michigan. Open symbols denote data from the CNR INM cavitation channel. The asymptotic increase demonstrates the increasing proportion of system inertia contributed by the fluid. All data were collected in quiescent fluid.

Figure 18. Ratio of generalized fluid mass to generalized structural mass as function of $Fn_{h}$ in FW flow ($0^{\circ }\leqslant \unicode[STIX]{x1D6FC}\leqslant 12^{\circ }$), FV flow ($7^{\circ }\leqslant \unicode[STIX]{x1D6FC}\leqslant 12^{\circ }$) and PC flow ($\unicode[STIX]{x1D6FC}=7^{\circ },12^{\circ }$ and $0.42\leqslant \unicode[STIX]{x1D70E}_{v}\leqslant 0.85$). All data are for $AR_{h}=1.0$. Mode 4 (lead–lag vibration) was not measured. Added-mass values in the FV regime are uniformly smaller than those in the FW regime, with PC data generally falling between the two. This is consistent with the displacement of heavy water with a gas-filled cavity that grows with increasing speed.

Interestingly, the fluid inertia dominates the structural inertia at all non-zero values of $AR_{h}$ for mode 1, and fluid inertia dominates for all modes for $AR_{h}>1.8$. For comparison, De La Torre et al. (Reference De La Torre, Escaler, Egusquiza and Farhat2013) found that the added-mass ratio for a fully submerged hydrofoil in fully wetted conditions reached approximately 2.5 for the first bending, which is consistent with the present data if two key factors are considered. The hydrofoil used in the study by De La Torre et al. possessed a geometric aspect ratio of 1.5 (as opposed to 3.45 in the present study), so the quantity of entrained mass was likely smaller for a given mode shape. The hydrofoil was also constructed from aluminium, so the structural modal mass was comparatively greater than a lighter weight material like PVC.

5.3.2 Effects of varying flow speed and flow regime

Figure 18 shows the added mass of the hydrofoil in FW, FV and PC flows. It should be noted that the added mass in FW flow is constrained by the assumption in § 5.1 to be equal to that in a quiescent fluid, i.e. independent of $Fn_{h}$. In PC and FV flows, there is a clear decreasing trend in the fluid’s inertial force with increasing speed. This may be explained by the behaviours in cavity sizes. In both PC and FV flows, an increase in the flow speed elicits larger cavities, which in turn displace increasing quantities of the dense water with light air or water vapour. The mean mass loading on the hydrofoil is reduced as a result. The added mass is uniformly larger in the PC regime than in the FV regime, which reflects the much-smaller cavities in the former regime (less than 40 % of the chord length).

Figure 19. Added-mass ratio plotted as a function of the cavitation parameter $\unicode[STIX]{x1D70E}_{c}/\unicode[STIX]{x1D6FC}$. Dashed lines represent (5.12) with coefficients in table 4. Data in the FW, PC and FV fall along contiguous sections of the same curve for each mode of vibration. The added-mass coefficient and cavitation parameter decrease progressively from FW to PC to FV flow regimes, showing that the fluid inertia is reduced as increasingly large gaseous cavities displace some of the surrounding liquid. Horizontal bars in the plots on the right indicate the propagated effects of the fluctuating cross-flow angle in the CNR INM channel ($\pm$ one standard deviation in the instantaneous cross-flow angle).

As in § 5.3.1, the added-mass ratios in each regimes is largest for mode 1 and decreases progressively with ascending mode orders. The relative decrease in added-mass ratios with the onset of PC or FV flow is also most substantial for low-order modes. Both observations further support the conjecture that the relative importance of the fluid inertia is tied closely to the swept volume of the mode shape. The cavity size is also affected by the vaporous cavitation number ($\unicode[STIX]{x1D70E}_{v}$) in PC flow and by the angle of attack ($\unicode[STIX]{x1D6FC}$) in both PC and FV flow. All values of $\unicode[STIX]{x1D70E}_{v}$ and $\unicode[STIX]{x1D6FC}$ are aggregated in figure 18, however, so their effects are not apparent.

5.3.3 Correlation with cavitation parameter

The added-mass ratios in PC and FV flows are proposed to be related to the physical sizes of the attendant cavities, so the cavitation parameter, $\unicode[STIX]{x1D70E}_{c}/\unicode[STIX]{x1D6FC}$, is a sensible choice for a correlation variable. The added-mass ratios are plotted in figure 19 as functions of $\unicode[STIX]{x1D70E}_{c}/\unicode[STIX]{x1D6FC}$ with horizontal uncertainty bars added to the magnified plots on the right, indicating the propagated effect of the fluctuating cross-flow angle in the cavitation channel (see Part 1 for detailed discussion). Equation (5.10) (modes 1 and 5) and equation (5.11) (modes 2 and 3) were used to model the hydrodynamic stiffness ratio term in (5.4). The new choice of independent correlation variable removes the overlap of the three regimes and produces a well-defined curve for each normal mode. Insets along the top of the figure illustrate the relative size of (or lack of) the cavity in each case.

Table 4. Fitted coefficients of the proposed correlation for the added-mass ratio (5.12).

The added mass will diminish with the growth of a gaseous cavity because, as previously stated, the growth of a cavity displaces a dense fluid with a light one. Classically, the length of a cavity on a given lifting section obeys an inverse power-law relationship with $\unicode[STIX]{x1D70E}_{c}/\unicode[STIX]{x1D6FC}$. In the limit of an infinite cavitation parameter, the added-mass ratio should be that identified in FW flow. As the cavitation parameter approaches zero (corresponding to a super-cavity), the added-mass ratio is expected to asymptotically approach zero as liquid water is increasingly displaced by light vapour. This leads to a proposed correlation of the form,

(5.12)$$\begin{eqnarray}\frac{M_{f,n}^{m}}{M_{s,n}^{m}}=\frac{\displaystyle A_{1}\left(\frac{\unicode[STIX]{x1D70E}_{c}}{\unicode[STIX]{x1D6FC}}\right)}{\displaystyle B_{1}\left(\frac{\unicode[STIX]{x1D70E}_{c}}{\unicode[STIX]{x1D6FC}}\right)+B_{0}}.\end{eqnarray}$$

The fitted values of the coefficients are given in table 4 and the resulting curves are overlaid on the respective plots of figure 19. The fits are quite good, notwithstanding some experimental scatter. It should be noted that the definition of the cavitation parameter at the mid-span of the immersed section is an important step, as that location generally corresponds with the locations of the mean cavity length in FV flow, the maximum cavity length in PC flow and the maximum sectional lift in all flow regimes Harwood et al. (Reference Harwood, Young and Ceccio2016c).

5.4.1 Effects of varying immersion depth in quiescent water

Generalized fluid to structural damping ratios are plotted in figure 20 as a function of $AR_{h}$. In terms of the damping groups proposed in § 4.5.1, the quantities in figure 20 reflect the modal quiescent fluid damping forces, relative to the modal structural damping forces. Data from the CNR INM cavitation channel and the University of Michigan at $AR_{h}=1.0$ once again compare favourably.

There exists a general increasing trend with increasing immersion depth – although a non-monotonic one – indicating that the quiescent fluid damping group increases with the proportion of the hydrofoil immersed in water. Four damping mechanisms are expected to occur in a quiescent fluid that do not occur in dry vibration: radiation of surface waves, acoustic radiation, acoustically coupled facility losses and viscous dissipation of eddies created by the hydrofoil’s motion. Acoustically coupled facility losses are not expected to be significant because no parasitic vibrations of either test environment were observed in partial immersion that were not observed during dry testing. Acoustic radiation damping is expected to be negligible for the same reason and because the flow occurs at extremely low Mach numbers. Potential flow theory suggests that inviscid wave radiation should be small at zero speed – supported by the favourable agreement between the results in the water-filled drum at Michigan (small domain for radiation) and the CNR INM cavitation channel (large domain). Thus, the increasing trend in generalized damping ratios is attributed to the viscous damping, which depends upon the amplitude and frequency of vibration, as well as the proportional immersion of the structure, as discussed in § 4.5.1. The results for mode 1 indicate that the fluid damping force is greater than the structural damping force across the range of $AR_{h}$ – a very different conclusion than that reached by considering the values of $\unicode[STIX]{x1D709}_{e}$ only. Modes 2, 3 and especially 5 imply a more equitable balance between the fluid and structural damping groups.

Figure 20. Ratio of generalized fluid damping to generalized structural damping for modes 1, 2, 3 and 5 as functions of $AR_{h}$. All data were collected in quiescent fluid. Damping forces for all modes increase with increasing immersion depth, probably as a result of increased viscous dissipation. For $AR_{h}\geqslant 1.7$, fluid damping forces become more important than structural damping forces for all modes.

Figure 21. Ratio of generalized fluid damping to generalized structural damping as function of $Fn_{h}$ in FW flow ($0^{\circ }\leqslant \unicode[STIX]{x1D6FC}\leqslant 12^{\circ }$), FV flow ($7^{\circ }\leqslant \unicode[STIX]{x1D6FC}\leqslant 12^{\circ }$) and PC flow ($\unicode[STIX]{x1D6FC}=7^{\circ },12^{\circ }$ and $0.42\leqslant \unicode[STIX]{x1D70E}_{v}\leqslant 0.85$). All data are for $AR_{h}=1.0$. There is a general increase in the hydrodynamic damping with increasing forward speed, particularly for FW flow. Hydrodynamic damping is dominant over the structural damping for modes 1 and 2.

5.4.2 Effects of varying flow speed and flow regime

Ratios of fluid to structural generalized damping forces are shown in figure 21. The trends are qualitatively similar to those of $\unicode[STIX]{x1D709}_{e}$ in figure 14 for each mode individually. As $Fn_{h}$ increases, the generalized damping ratios increase with the inclusion of the hydrodynamic damping group from § 4.5.1. The trend in hydrodynamic damping as $Fn_{h}$ increases is consistent with (4.22). This observation extends to modes 1, 2 and 3, while mode 5 shows very little dependence upon flow speed. The summed quiescent and hydrodynamic damping for modes 1 and 2 reach maximum values approaching ten times the maximum values of quiescent fluid damping alone. This highlights the dominant role of wave radiation damping, inviscid wake damping and the increase of viscous damping with forward speed. The inviscid wake damping is expected to increase linearly with increasing speed (although this behaviour assumes a fixed reduced frequency).

For mode 1, the FV damping is nearly identical to the FW damping, while the PC damping is substantially smaller. As described previously, however, the quality of the extracted damping parameters for mode 1 is suspect for $Fn_{h}\geqslant 2.0$. Mode 2 shows a reduction in damping with the onset of ventilated or cavitating flow – again probably due to a reduction in the circulatory wake damping and viscous damping terms. Modes 3 and 5 demonstrate increased damping with the onset of multiphase flows. In both cases, this is thought to occur because the wake, viscous and radiation damping terms are small to begin with – and are thus less subject to attenuation – and because the higher frequency vibrations of these modes increase the deposition of energy into the cavity interfaces, where it is dissipated via cavity shedding and re-entrant flow. In fact, the transfer of mechanical energy from higher-frequency modes of vibration into the cavity interface leads to possible means of controlling multiphase flow, as described in § 6.1.

5.4.3 Correlation with reduced frequency

Figure 22 shows the ratio of generalized hydrodynamic damping to structural damping plotted against the reduced resonant frequency $k=\unicode[STIX]{x1D714}_{0}c/2U$. The hydrodynamic damping was estimated by subtracting the quiescent fluid damping from each case. Significant scatter is still present in the data, but a clear trend emerges, indicating a peak in the fluid damping at $k\approx 0.3$. The peak and values to its left may be attributed in part to the spurious reduction in damping for mode 1 at higher speeds (low reduced frequencies) observed in § 5.4.1. However, for the data to the right of the peak, there is a uniform decrease in the hydrodynamic damping with increasing reduced frequency – an observation shared with Blake & Maga (Reference Blake and Maga1975). This highlights the presence of the inviscid wake damping in (4.22), which is a function of $k$. Scatter may be attributed to the other terms present in the damping group.

Figure 22. Generalized damping ratio plotted as a function of the reduced frequency $k=\unicode[STIX]{x1D714}_{0}c/2U$ at resonance. Despite significant scatter, the data from modes 1, 2, 3 and 5 for all three flow regimes appear to describe a curve with a maximum near $k\approx 0.3$, which may signal a maximum in circulatory lift opposing structural velocities. The spuriously low damping estimates for mode 1 are conflated with this result.