2.1 Experimental model and fixture

The model used in the experiments possesses a uniform semi-ogival section along the span, with a 27.94 cm chord and a maximum thickness of 2.79 cm, a circular-arc (ogival) forebody with a radius of curvature of 84.84 cm, and a rectangular afterbody. The section has a leading edge radius of $0.01c$ and a blunt trailing edge, as shown in figure 4. The hydrofoil has a rectangular planform with a 91 cm span, and was machined as a single piece from 6061 aluminium. The geometry resembles that used in past studies, particularly the models of Waid (Reference Waid1968), Rothblum et al. (Reference Rothblum, Mayer and Wilburn1969) and Swales et al. (Reference Swales, Wright, McGregor and Cole1973, Reference Swales, Wright, McGregor and Rothblum1974).

Figure 4. Cross-section of the hydrofoil. The model was machined from 6061 aluminium.

The hydrofoil was clamped at the root and slung vertically in a cantilevered configuration to pierce the water free surface, as shown in figure 5. A steel fixture frame was mounted between two longitudinal structural members of the towing-tank carriage (figure 5 b). Threaded feet were used to bring the frame to within $\pm 0.1^{\circ }$ of level, as measured in the $X$ – $Z$ and $Y$ – $Z$ planes. The hydrofoil was coupled to the frame by way of a load cell that, in turn, attached to a drum that rotated about a vertical axis, permitting the yaw angle (angle of attack) to be varied. An ATI Omega-190 US 800-6000 six-degree-of-freedom force–torque transducer was used to measure forces and moments at the root of the hydrofoil.

Figure 5. Experimental apparatus for the surface-piercing foil experiments. (a) The foil geometry, foil dimensions and coordinate system used in the following discussion. (b) An exploded view of the components comprising the experimental rig. The entire apparatus attaches to the towing-tank carriage.

2.2 Videography and surface-flow visualization

Videography was performed using GoPro^® brand cameras (HERO3 Black^® models) in waterproof housings. Video was recorded with a resolution of $1920\times 1080$ pixels at a rate of 60 frames  $\text{s}^{-1}$ . Lighting was provided by commercial halogen work lamps mounted below the carriage and aimed at the hydrofoil. Video streams were recorded above and below the water and were collectively synchronized with the data collection by an LED indicator in the field of view of each camera.

A limited number of surface-flow visualizations were conducted using a grid of paint dots, following the method of Green (Reference Green1988) and Green & Acosta (Reference Green and Acosta1991). Eight parts yellow artist’s oil paint and one part white oil primer were mixed with five to twelve parts refined linseed oil to achieve a range of shear-thinning behaviours, suitable for trials conducted at different speeds. Droplets of volume $10~{\rm\mu}\text{L}$ were applied as a grid on the suction surface of the hydrofoil with a pipette, with a spacing of approximately 1–2.5 cm (0.5–1 in.), totalling between 50 and 150 dots. Between each trial, the foil was wiped clean and the dots were reapplied.

2.3 Experimental procedure

The yaw angle, or angle of attack ( ${\it\alpha}$ ), was measured as the rotation of the hydrofoil in the $X{-}Y$ plane. The yaw angle ${\it\alpha}$ was set by visually sighting one-degree marks machined into the yaw drum (see figure 5) against a reference line calibrated to match the longitudinal axis of the towing tank. The drum was secured by tightening a collar clamp. The depth of immersion was varied by adding and draining water from the towing tank and visually sighting the water line against tick marks drawn along the span of the hydrofoil. Speed was measured by an optical encoder wheel, and instantaneous carriage speed was governed by an on-board controller. The variable $U_{\infty }$ denotes the steady-state speed entered as an input to the carriage speed controller, while $u(t)$ is the instantaneous velocity of the carriage, measured by the optical encoder. During a run, the carriage accelerated to the specified speed ( $u(t)=U_{\infty }$ ) and decelerated again along a programmed speed profile. The acceleration of the carriage during the ramp-up and ramp-down stages was software-limited to $\pm 0.025$ – $0.525~\text{m}~\text{s}^{-2}$ . The ranges of parameters investigated are given in table 2. Trials were conducted by varying $U_{\infty }$ for each configuration of ${\it\alpha}$ and $AR_{h}$ , then varying ${\it\alpha}$ within each fixed value of $AR_{h}$ , and varying $AR_{h}$ once all values of ${\it\alpha}$ and $Fn_{h}$ had been tested.

Table 2. Model dimensions and ranges of variables used in the present experiment.

For each trial, data collection was started while the carriage was at rest. For steady-state data collection, ramp-up and ramp-down times were minimized to increase the amount of data collected at the target condition of $u(t)=U_{\infty }$ . To study formation, elimination and other transient processes, the ramp-up and/or ramp-down times were typically made longer to minimize inertial effects. A small-diameter air jet (driven by a pneumatic tank at 350–700 kPa) was used to perturb the flow at the junction of the free surface and the foil leading edge in order to trigger ventilation at small yaw angles. Airflow through the jet was actuated by a solenoid valve, in turn controlled by an output from the data acquisition computer. The valve was set to open when $u(t)$ remained within $\pm 3\,\%$ of $U_{\infty }$ continuously for a user-specified period of time, and to remain open for a user-specified duration. The duration of air injection was varied between 0.25 and 1 s as needed to achieve ventilation. Sufficient time (5–15 min) was permitted between runs to allow waves and currents in the tank to dissipate.

2.4 Data collection and processing

The speed and load-cell outputs were recorded using a pair of 16-bit National Instruments multifunction DAQ cards on either a PCI-e bus or a dedicated PXI chassis. All signals were sampled at 1000 Hz, with asynchronism of less than $1~{\rm\mu}\text{s}$ introduced by channel muxing on the analogue-to-digital converters (ADCs). All voltage signals were normalized to a $\pm 10$ volt range to maximize the dynamic range of the ADCs.

Windows around steady-state data were manually specified. In some cases, a short window was taken on data recorded during acceleration or deceleration phases. Data taken during transient flows were considered to be suitably quasi-steady when no significant topological changes were observed and when all recorded speeds remained within $\pm 3\,\%$ of the desired steady velocity during the window in question. In other words, a time window $t_{w}$ was specified as $t_{w}=\{t\mid u(t)~\text{within}~\pm 3\,\%~\text{of}~U_{\infty }\}$ . In this way, slowly varying ramped runs were used to canvas several quasi-steady flow conditions in a single run. Statistics for speed, force and moment coefficients, instantaneous accelerations, and other inferred quantities were calculated on these windows. Some transient effects, due to fluid inertia, are unavoidable. However, for $Fn_{h}>1$ , the effect of unsteady inertial fluid loading on measured lift and drag may be neglected. The justification for this statement can be found in appendix B. The nominal uncertainty bounds of measured quantities and the propagated uncertainty in results of interest are presented in table 3.

Table 3. Nominal error bounds of measured quantities are given in the first section. Values are quoted from the load-cell manufacturer in the case of forces and moments, estimated from recorded data in the case of speed, and taken to be one quarter of the smallest delineation for yaw angle and tip immersion. It should be noted that force/torque uncertainties are quoted at 50 % of the rated load, and include cumulative noise and quantization error. Propagated uncertainty bounds on quantities of interest are given in the second part. It should be noted that different values are given, corresponding to the three immersed aspect ratios that were tested.

3.1 The re-entrant jet at cavity closure

Where a cavity closes on a solid surface, streamlines inside the stagnation streamline form a jet spray that undercuts the cavity, known as a re-entrant jet (Laberteaux & Ceccio Reference Laberteaux and Ceccio2001a ). In 2D flows, the re-entrant jet may impinge on the upstream cavity boundary, causing ‘pinch-off’. A well-known example is the phenomenon of sheet–cloud cavitation, which occurs when a re-entrant jet induces periodic shedding of vaporous partial cavities. When the line of cavity closure is non-normal to the incoming flow, the re-entrant jet is a reflection of the incoming flow about the cavity closure line, as shown in figure 6 (De Lange & De Bruin Reference De Lange and De Bruin1998; Laberteaux & Ceccio Reference Laberteaux and Ceccio2001b ; Franc & Michel Reference Franc and Michel2006). If incoming flow encounters a cavity closure line at a local angle of ${\it\Phi}$ from the horizontal plane, the jet flow will be directed at an angle of $2{\it\Phi}$ , as shown by the arrows in figure 6. At a given depth beneath the free surface, $z^{\prime }$ , the magnitude of the jet velocity vector ( $\boldsymbol{U}_{j}$ ) may be obtained by applying Bernoulli’s equation between the upstream flow (known velocity) and the cavity boundary (known pressure) to obtain

(3.1) $$\begin{eqnarray}\Vert \boldsymbol{U}_{j}(z^{\prime })\Vert =u\sqrt{1+{\it\sigma}_{c}(z^{\prime })}.\end{eqnarray}$$

The respective streamwise ( $U_{j}$ ) and spanwise ( $W_{j}$ ) components of the re-entrant jet velocity are (Harwood et al. Reference Harwood, Brucker, Miguel, Young and Ceccio2014)

(3.2a ) $$\begin{eqnarray}\displaystyle & \displaystyle U_{j}(z^{\prime })=U_{0}\sqrt{1+{\it\sigma}_{c}(z^{\prime })}\cos (2{\it\Phi}(z^{\prime })), & \displaystyle\end{eqnarray}$$

(3.2b ) $$\begin{eqnarray}\displaystyle & \displaystyle W_{j}(z^{\prime })=U_{0}\sqrt{1+{\it\sigma}_{c}(z^{\prime })}\sin (2{\it\Phi}(z^{\prime })). & \displaystyle\end{eqnarray}$$

${\it\alpha}\geqslant 5^{\circ }$

Figure 6. Schematic representation of the re-entrant jet on a spanwise-varying cavity. When incoming flow (arrows) encounters an oblique line of cavity closure at a local angle ${\it\Phi}$ , the flow is reflected about the normal to the cavity closure line so that the resulting jet spray is directed along an angle of $2{\it\Phi}$ from the horizontal plane.

The notion of re-entrant as a reflection of incoming flow is reinforced by the surface-flow visualization shown in figure 7. Paint dots were applied as described in § 2.2, and the hydrofoil was accelerated along a ramped velocity profile at a yaw angle sufficiently large for a ventilated cavity to spontaneously develop ( ${\it\alpha}=14^{\circ }$ ). As the forward velocity, $u$ , increased, the ventilated cavity grew in size. The re-entrant jet at the cavity closure line was swept over the paint dots, causing the dots to shear in the direction of the local jet spray. Some smearing of the dots was unavoidable, as the dots were sheared somewhat in the streamwise direction prior to their interaction with the re-entrant jet; nevertheless, the primary directions of streaking clearly coincide with the superimposed arrows, which indicate the local $2{\it\Phi}$ trajectories at their intersections with the respective cavity profiles.

Figure 7. Paint streaks on the suction side of a hydrofoil in accelerating flow at ${\it\alpha}=14^{\circ }$ , $AR_{h}=1$ . As the speed increased, the cavity grew (see b–d), sweeping the re-entrant jet over the surface of the foil and shearing the paint in the direction of the jet. Cavity profiles are drawn for $Fn_{h}=1.15,1.5$ and 2.0. Superimposed on each cavity profile are arrows indicating the local theoretical re-entrant jet direction of $2{\it\Phi}$ . The streaks in the paint coincide with the theoretical jet directions, indicating that the theory ((3.2a ) and (3.2b )) correctly predicts the trajectory of the re-entrant flow.

When the local angle of the cavity closure line is ${\it\Phi}>45^{\circ }$ , the re-entrant jet will possess an upstream velocity component. With enough momentum, the jet impinges on the leading edge of the ventilated cavity, causing a large-scale shedding of the cavity via pinch-off (as in the case of sheet–cloud cavitation, described earlier). If this occurs over a significant portion of the span, then a ventilated cavity becomes unstable.

At this point, it is useful to define stability in the context of ventilated flows. If a configuration, or regime, of the flow remains unchanged following any perturbation to the flow, then that regime is considered to be exclusively and globally stable. If small perturbations do not affect the flow regime, but large perturbations cause transition to another flow regime, then the regime is considered to be locally stable. The coexistence of a locally stable regime and another locally or globally stable regime constitutes a bi-stable set of conditions. For given flow conditions, a flow regime is considered to be unstable if it cannot be sustained, even if left unperturbed.

A criterion of $\bar{{\it\Phi}}=45^{\circ }$ is proposed on the stability of fully ventilated cavities, where $\bar{{\it\Phi}}$ is the angle of a linear approximation to the cavity closure line. To verify that $\bar{{\it\Phi}}$ is a suitable metric, images were taken of ventilated cavities in decelerating flows, immediately preceding the instant when the cavity became unstable. The cavity length was measured at three spanwise stations, as shown in figure 8(a), through which was fitted an affine curve to get the value of $\bar{{\it\Phi}}$ , measured from the horizontal plane. The criterion $\bar{{\it\Phi}}$ is plotted as a function of ${\it\alpha}$ and $AR_{h}$ in figure 8(b). Measurements were translated into real-world coordinates by using an inverse projective transform. The results show the cavity angle at washout to be tightly clustered around a mean of $\bar{{\it\Phi}}\approx 41^{\circ }$ for all values of ${\it\alpha}$ and $AR_{h}$ , which is acceptably close to the proposed stability boundary at $\bar{{\it\Phi}}=45^{\circ }$ . With this stability condition, the topology of a cavity may be used to unambiguously distinguish partial ventilation from full ventilation in 3D flows.

Figure 8. Experimental quantification of cavity closure angles at the point of cavity instability via pinch-off for immersed aspect ratios of $AR_{h}=0.5$ , 1 and 1.5. The example shown in (a) is at $Fn_{h}=1.5$ , ${\it\alpha}=20^{\circ }$ , $AR_{h}=1.0$ . Cavity closure line profiles were linearly approximated to find $\bar{{\it\Phi}}$ at the instant of cavity instability. (b) Angles of cavity closure lines as a function of and $AR_{h}$ . The mean value of $\bar{{\it\Phi}}$ is $40.75^{\circ }$ and the dashed line corresponds to the proposed stability criterion of $45^{\circ }$ .

3.2 Flow-regime criteria

Fully wetted (FW) flow is defined as the regime in which no pronounced entrainment of gas occurs. In the present experiments, a small aerated base cavity was typically observed in the separated wake of the blunt trailing edge. However, the effects of the trailing edge were confined to a relatively small region immediately aft of the trailing edge; neither the flow separation nor the base cavity materially affected the hydrodynamic response of the hydrofoil (Perry Reference Perry1955). An example of FW flow is shown in figure 9. The nominal free surface indicates the initial calm water surface identified by the $z^{\prime }=0$ plane, where it intersects with the suction surface of the foil. The free-surface profile changes when subjected to the pressure distribution around the hydrofoil, and is termed the ‘deformed’ free surface.

Figure 9. Fully wetted flow at ${\it\alpha}=10^{\circ }$ , $Fn_{h}=2$ , $AR_{h}=1.0$ . The FW regime shows no entrainment of air along the suction surface ( $D=0$ , as measured from the nominal free surface). Some base ventilation occurs aft of the blunt trailing edge of the foil.

Fully ventilated (FV) flow is defined as the case where a cavity is entrained along the entire immersed span of the hydrofoil suction surface ( $D=h$ , where $D$ is the maximum depth of the ventilated cavity beneath the nominal free surface), and where no inherently destabilizing re-entrant jet occurs ( $\bar{{\it\Phi}}<45^{\circ }$ ). Figure 10 shows an example of a fully ventilated cavity. The suction surface of the hydrofoil (pictured) is enveloped entirely inside a ventilated cavity. The near wall of the cavity detaches from the leading edge of the hydrofoil. The pressure side of the hydrofoil is fully wetted, and flow detaches smoothly from the corner of the pressure side and trailing edge to form the opposing cavity wall. Large spray sheets develop on both sides of the hydrofoil; the lower edge of the suction-side spray sheet is visible in figure 10. It should be noted that the angle of the cavity closure line is very shallow ( $\bar{{\it\Phi}}\ll 45^{\circ }$ ). As a result, the streamwise component of any re-entrant flow is directed towards the trailing edge of the hydrofoil, where it presents no detriment to the cavity stability. In figure 10, a strong tip vortex is present, the core of which aerates by ingesting air from the cavity. When the flow is in a fully ventilated state, the blunt trailing edge is contained entirely within the cavity, so it does not interact with the liquid flow. In the case pictured, and in other cases with a sufficiently strong tip vortex, the circumferential flow around the vortex impinges on the near (as pictured) cavity wall, forcing a small jet horizontally across the cavity. The impact of this jet on the far cavity wall causes a vertical jet spray to develop in the ventilated wake.

Figure 10. Fully ventilated flow at ${\it\alpha}=15^{\circ }$ , $Fn_{h}=3.0$ , $AR_{h}=1.0$ . The suction surface (pictured) is contained entirely within the walls of the cavity. The cavity satisfies both FV flow criteria ( $D=h$ and $\bar{{\it\Phi}}\leqslant 45^{\circ }$ ). In the case pictured, a strong tip vortex is present, the core of which aerates by ingesting air from the cavity.

Partially ventilated (PV) flow is defined as the entrainment of a cavity which either does not span the entire immersed portion of the hydrofoil or does not meet the stability conditions to sustain fully ventilated flow (e.g. a destabilizing re-entrant jet exists). Figure 11 depicts two examples of partial ventilation. The flow in figure 11(a) is designated PV because the cavity does not reach the immersed tip of the hydrofoil ( $D<h$ ). In this case, the cavity depth and length are potentially unsteady. The flow pictured in figure 11(b) is so designated because the approximate cavity closure line, shown as a solid line fitted through the cavity closure region, exceeds the $\bar{{\it\Phi}}=45^{\circ }$ stability condition, which is indicated by the dashed white line. As a result, a significant portion of the re-entrant jet is directed towards the leading edge. At small angles of attack, the resulting upstream flow may cause unsteady shedding of bubbly structures in a region confined to the trailing edge of the cavity, while at larger angles of attack, the jet destabilizes the entire cavity by impinging on the cavity interface and leading to pinch-off. In either case, the increased unsteadiness and decreased stability of the flow when $\bar{{\it\Phi}}\geqslant 45^{\circ }$ serve to distinguish it from FV flow.

Figure 11. Partially ventilated flow, with (a)  ${\it\alpha}=20^{\circ }$ , $Fn_{h}=1.0$ , $AR_{h}=1.0$ ; (b)  ${\it\alpha}=7.5^{\circ }$ , $Fn_{h}=1.54$ , $AR_{h}=1.0$ . (a) The case where the cavity does not reach the free tip ( $0<D<h$ ). (b) The case where the cavity reaches the free tip ( $D=h$ ), but the angle of the solid line fitted through the cavity closure region is greater than the critical angle ( $\bar{{\it\Phi}}=45^{\circ }$ , shown as a dashed white line).

The criteria for the three flow regimes may be summarized as follows:

$D=0,\quad \text{fully wetted (FW) flow}$ ;

$D=h\quad \text{and}\quad \bar{{\it\Phi}}<45^{\circ },\quad \text{fully ventilated (FV) flow}$ ;

$0<D<h\quad \text{or}\quad \bar{{\it\Phi}}\geqslant 45^{\circ },\quad \text{partially ventilated (PV) flow}$ .

Here, $D$ is the depth of a ventilated cavity, measured from the free surface, and $\bar{{\it\Phi}}$ is the approximate angle of the cavity closure line.

3.3 Ventilation and hydrodynamic loading

When ventilation occurs, most or all of a lifting-surface suction surface becomes encapsulated in a cavity. The pressure inside the cavity after ventilation occurs is, by necessity, higher than the local absolute pressure in the prior wetted flow. Suction pressures along the portions of the hydrofoil within the cavity are thus strongly attenuated. As a result, ventilation can induce dramatic reductions in realizable lift, as observed by Breslin & Skalak (Reference Breslin and Skalak1959), Rothblum et al. (Reference Rothblum, Mayer and Wilburn1969) and Swales et al. (Reference Swales, Wright, McGregor and Rothblum1974). The 3D lift, drag and yawing moment coefficients are defined as

(3.3a ) $$\begin{eqnarray}\displaystyle & \displaystyle C_{L}=\frac{F_{Y}}{1/2{\it\rho}u^{2}hc}, & \displaystyle\end{eqnarray}$$

(3.3b ) $$\begin{eqnarray}\displaystyle & \displaystyle C_{D}=\frac{-F_{X}}{1/2{\it\rho}u^{2}hc}, & \displaystyle\end{eqnarray}$$

(3.3c ) $$\begin{eqnarray}\displaystyle & \displaystyle C_{M}=\frac{M_{Z}}{1/2{\it\rho}u^{2}hc^{2}}. & \displaystyle\end{eqnarray}$$

$F_{X}$

$F_{Y}$

$M_{Z}$

Figure 12 shows $C_{L}$ in the FW and FV flow regimes; a distinctive grouping of the data into the two regimes is evident. The PV data are omitted for clarity, but may be found in appendix A. The lift in the FV regime is significantly lower than that in the FW regime, and the difference in lift between the two regimes tends to increase with increasing ${\it\alpha}$ . It should be noted that a line fitted to the FV data will have an apparent non-zero intercept because the profile of the foil pressure surface and the streamline bounding the cavity together present to the flow a cambered lifting surface (Breslin & Skalak Reference Breslin and Skalak1959).

The nonlinear lifting-line model from § 1.2 was also used to approximate the lift. In the FV regime, the root-finding approach of § 1.2.2 was used to resolve the distributions of $L_{c}/c$ , $a_{0}$ , ${\it\alpha}_{2D}$ and $C_{l_{2D}}$ iteratively. In the FW regime, equation (1.7) was replaced with $L_{c}/c=0$ and (1.9) was replaced with $a_{0}=2{\rm\pi}$ . In both regimes, the lifting-line model correctly captures the trends of the experimental measurements.

Figure 12. The 3D lift coefficient as a function of ${\it\alpha}$ at $Fn_{h}=2.5$ and $AR_{h}=1.0$ . Propagated uncertainties from table 3 are shown as grey bars. Black bars are used to denote the standard deviation of $C_{L}$ within the respective steady-state time window. Where duplicate points exist, the black bars correspond to the standard deviation of the estimate of the mean $C_{L}$ . Between the bifurcation angle ( ${\it\alpha}_{b}$ ) and the stall angle ( ${\it\alpha}_{s}$ ), both wetted and ventilated flow regimes were observed, signalling a bi-stable range of ${\it\alpha}$ . The nonlinear lifting-line model calculations are also shown, with thin lines used to indicate predictions outside the experimentally observed ranges of ${\it\alpha}$ .

Figure 13. The 3D drag coefficients as a function of ${\it\alpha}$ at $Fn_{h}=2.5$ and $AR_{h}=1.0$ . Both regimes exhibit a classical quadratic variation of $C_{D}$ with ${\it\alpha}$ . The drag coefficients are nearly continuous between the two regimes. The results of the lifting-line model include induced and frictional drag only, suggesting that the increased profile drag and spray drag of the cavity roughly negate the reduction in frictional and induced drag.

The drag coefficients are shown in figure 13. The dashed lines indicate the summed frictional and lift-induced drag coefficients from the lifting-line model. The skin friction coefficient was estimated via the ITTC 1957 correlation line (Larsson & Raven Reference Larsson, Raven and Paulling2010) to be $C_{D_{f}}=0.0047$ at the conditions in figure 13. The skin friction was integrated over both surfaces of the foil in the FW regime and over the pressure surface only in the FV regime. Breslin & Skalak (Reference Breslin and Skalak1959) posited that, for certain geometries, a ventilated cavity yields an increased profile drag that roughly offsets the reduction in skin friction and induced drag. The frictional drag represents only a very minor contribution to the total drag in both regimes, so the near continuity of the measured total drag between the FW and FV flow regimes implies that any changes in the remaining components of drag (induced drag, profile drag, spray drag, etc.) approximately negate one another. The significant reduction in the induced drag predicted by the lifting-line model is caused by the previously noted decrease in lift, and suggests that those drag components not captured by the lifting-line model, namely profile and spray drag, are substantially increased by the formation of a ventilated cavity.

The yawing moment coefficient, $C_{M}$ , measured about mid-chord, is shown in figure 14. The FW data follow a roughly linear trend, except near the stall boundary, where a reduction in the yawing moment indicates impending stall and an accompanying shift in the centre of pressure towards the mid-chord. The FV data exhibit a significant reduction in $C_{M}$ , compared with the wetted data. This is partly due to the reduction in $C_{L}$ shown in figure 12, and is compounded by the movement of the centre of pressure towards the mid-chord, resulting in a sub-linear trend in $C_{M}$ with respect to changing ${\it\alpha}$ . The lifting-line code was used to approximate the yawing moment, also shown in figure 14. The dimensionless location of the centre of pressure for FW sections is estimated to be $e_{FW}=x_{cp}/c=1/4$ , where $x_{cp}$ is the location of the centre of pressure forward of mid-chord. For supercavitating sections, $e_{SC}=3/16$ (Akcabay & Young Reference Akcabay and Young2014). A sigmoid function was used to smoothly interpolate between the bounding values as a function of sectional cavity length,

(3.4) $$\begin{eqnarray}e(L_{c}/c)=\frac{1}{2}\left\{e_{FW}\left[1-\tanh \left(\frac{L_{c}/c-0.5}{0.25}\right)\right]+e_{SC}\left[1+\tanh \left(\frac{L_{c}/c-0.5}{0.25}\right)\right]\right\}.\end{eqnarray}$$

Figure 14. The 3D yawing moment coefficient as a function of ${\it\alpha}$ at $Fn_{h}=2.5$ and $AR_{h}=1.0$ . The yawing moment is defined about mid-chord. It is strongly reduced in ventilated flow, indicating that the centre of pressure moves towards the mid-chord as the suction surface becomes aerated. The lifting-line model, in concert with (3.4), was used to estimate $C_{M}$ , plotted as dashed lines. The relatively good agreement suggests that the combined reductions in lift and shift in the centre of pressure together adequately describe the trend in $C_{M}$ .

Figure 15. The $C_{L}$ surface as a function of ${\it\alpha}$ and $Fn_{h}$ for $AR_{h}=1$ . Contours of constant $C_{L}$ are shown as solid lines in the FW and FV regimes (the PV surface and contours are omitted for clarity). It is a strong function of $Fn_{h}$ in the FV regime, where for a fixed ${\it\alpha}$ , $C_{L}$ decreases as $Fn_{h}$ increases. At low speeds ( $Fn_{h}\leqslant 1$ ), laminar separation occurs at the accompanying low Reynolds numbers ( $\mathit{Re}_{c}\leqslant 5\times 10^{5}$ ), causing an increase in $C_{L}$ in the FW regimes. Affine fits of $C_{L}$ as a function of ${\it\alpha}$ are shown as dotted lines at each $Fn_{h}$ , with the intersection between the FW and FV fits indicated by cross-hatched circles. The dashed lines are orthographic projections of the locus of intersection points onto the ${\it\alpha}$ – $Fn_{h}$ plane and the $Fn_{h}$ – $C_{L}$ plane. A heavy solid line indicates the locus of bifurcation angles ( ${\it\alpha}_{b}$ ), forming a bifurcation boundary, below which only FW flow was observed.

The effect of Froude number is also of interest. In figure 15, $C_{L}$ is plotted as a function of ${\it\alpha}$ at values of $Fn_{h}$ between 0.5 and 3.5, forming surfaces representing the FW and FV regimes. Two-dimensional plots of hydrodynamic load coefficients with varying Froude number may be found in figures 31–33 in appendix A. In the FW regime, the $C_{L}$ contours are parallel to the $Fn_{h}$ axis for $Fn_{h}>1$ , indicating that $C_{L}$ is speed-dependent only at very low speeds. For $Fn_{h}\leqslant 1$ , the Reynolds numbers are low ( $\mathit{Re}_{c}\leqslant 5\times 10^{5}$ ), resulting in leading edge laminar separation, which in turn causes a characteristic increase in lift (Breslin & Skalak Reference Breslin and Skalak1959). In the fully ventilated regime, lift monotonically decreases with increasing Froude number at a fixed angle of attack, made apparent by the curvature of the $C_{L}$ contours in the FV regime. This behaviour is due in part to longer cavity lengths at each section along the span (caused by the reduction in sectional ${\it\sigma}_{c}$ at high Froude numbers), which lead to reduced lift coefficients (see figure 2 b). Additionally, longer sectional cavity lengths present a smaller amount of apparent camber to the flow, causing further reduction in the sectional values of $C_{l_{2D}}$ . Affine curve fits through the $C_{L}$ values in the FW and FV flow regimes with respect to ${\it\alpha}$ are shown as dotted lines. Cross-hatched circular markers indicate the intersection between the FW and FV affine data fits at each respective value of $Fn_{h}$ . The dependence of $C_{L}$ upon $Fn_{h}$ in the FV regime leads to intersection points at increasingly large angles of attack as the Froude number is decreased. Orthographic projections of this locus are made onto the ${\it\alpha}$ – $Fn_{h}$ and $Fn_{h}$ – $C_{L}$ planes and are shown as black dashed lines in figure 15. It should also be noted that, while not shown in figure 15 to avoid overcrowding, the nonlinear lifting line correctly represents the qualitative behaviour of $C_{L}$ with changing $Fn_{h}$ . The model results do not yield a good qualitative match to experimental measurements across the entire $Fn_{h}$ domain – thought to be because effects such as the apparent camber are not modelled – but the decrease in $C_{L}$ with increasing $Fn_{h}$ in the FV regime is present.

Partially ventilated data are also present in figure 15; however, the $C_{L}$ contours and surface fit are omitted for the PV regime to avoid overcrowding. It should be recalled that the bifurcation angle ( ${\it\alpha}_{b}$ ) was shown in figure 12 to be ${\it\alpha}_{b}\approx 2.5^{\circ }$ at $Fn_{h}=2.5$ and $AR_{h}=1.0$ . With the inclusion of PV data, the meaning of the bifurcation angle may be further clarified. The bifurcation angle is defined for a given Froude number as the angle below which FW flow was the sole flow regime observed during experiments. In figure 15, the bifurcation angles form a ‘bifurcation boundary’, shown as a solid black line. The bifurcation boundary takes on a value of ${\it\alpha}_{b}\approx 2.5^{\circ }$ at $Fn_{h}\geqslant 2$ , and increases along the ${\it\alpha}$ -axis with decreasing values of $Fn_{h}$ for $Fn_{h}<2$ . In the neighbourhood of ${\it\alpha}_{b}$ , values of $C_{L}$ in the FW, FV and (where applicable) PV regimes are nearly coincident. This observation suggests that the suction-surface pressures of the wetted flow near ${\it\alpha}_{b}$ are only mildly sub-atmospheric, so that ventilation formation does not materially affect the magnitude of suction-surface pressures. Reinforcing this hypothesis, the intersection of the affine curve fits approximates the bifurcation angle ( ${\it\alpha}_{b}$ ) at each value of  $Fn_{h}$ .

The bifurcation is so named because above ${\it\alpha}_{b}$ , other alternate flow regimes become stable, forming characteristic branches in the plots of stable hydrodynamic load coefficients. The other critical angle is the stall angle ( ${\it\alpha}_{s}$ ), above which FW flow was never observed, and so named because large eddies were observed in the wake of the foil at ${\it\alpha}>{\it\alpha}_{s}$ , indicating massive flow separation. For the present foil and for $AR_{h}=1$ , stall was observed at approximately ${\it\alpha}_{s}=14^{\circ }{-}15^{\circ }$ for all values of $Fn_{h}$ tested. Smaller values of $AR_{h}$ were observed to increase the stall angle slightly (see appendix A). A range of bi-stable flow conditions exists for ${\it\alpha}_{b}\leqslant {\it\alpha}\leqslant {\it\alpha}_{s}$ in figures 12–14, as well as in the overlap of the FW and FV surfaces in figure 15. Under such conditions, stable FW flow overlaps with PV or FV flow, and the flow can take on alternate stable configurations.

Figure 16. Steady flow regime map for the hydrofoil as a function of ${\it\alpha}$ and $Fn_{h}$ at a fixed aspect ratio of $AR_{h}=1.0$ . Shaded regions indicate experimentally observed stable flow regimes. Overlapping zones (numbered 1 through 3) are denoted by hatching, and are indicative of bi-stable flow. Symbols indicate experimentally observed regime boundaries, through which are fitted approximate curves. Uncertainty bars indicate the range in which ventilation transition events were observed to occur. Heavy solid lines on the left and right indicate the bifurcation boundary (the locus of ${\it\alpha}_{b}$ , as shown in figure 15) and the stall boundary (the locus of maximum observed stall angles, ${\it\alpha}_{s}\approx 15^{\circ }$ ). The heavy dotted line is an orthographic projection of the locus of points marking the approximate intersection of $C_{L}$ in the FW and FV flow regimes, also shown in figure 15.

3.4 Stability regions of the three stable flow regimes

The FW, FV and PV flow regimes are mapped as functions of ${\it\alpha}$ and $Fn_{h}$ for an immersed aspect ratio of $AR_{h}=1.0$ in figure 16. The experimentally observed flow regimes reveal three distinct stability regions (indicated by solid shading), which overlap in zones 1, 2 and 3 (indicated by hatching). The bifurcation boundary shown in figure 15 is drawn as a heavy solid line in figure 16. Below and to the left of this boundary, FW flow alone was observed. It should be noted that ${\it\alpha}=2.5^{\circ }$ was the smallest non-zero angle tested; the verticality of the bifurcation boundary is therefore probably a result of the testing resolution, and FV or PV flow might be attainable at ${\it\alpha}<2.5^{\circ }$ for large $Fn_{h}$ . The rightmost extent of the FW regime is indicated by the vertical stall boundary, formed by the collection of stall angles ( ${\it\alpha}_{s}$ ), which appears to be independent of $Fn_{h}$ . The regions of overlap indicate bi-stable flow conditions, where mutually stable flow regimes coincide. For example, the bi-stable range of ${\it\alpha}$ shown in figures 12–14 corresponds to the overlap of FW and FV regions (in zone 1) along a horizontal cut through figure 16 at $Fn_{h}=2.5$ . In addition to overlap with respect to ${\it\alpha}$ , the three regimes overlap their neighbours in the $Fn_{h}$ direction as well, which has not previously been noted. The edges of the stability regions are populated with symbols, which denote experimentally observed processes that cause transition from one flow regime to another. These ventilation transition mechanisms will be described in § 4. The locus of intersection points between fitted values of $C_{L}$ in the FW and FV regimes (from figure 15) has been orthographically projected onto the ${\it\alpha}$ – $Fn_{h}$ plane, shown by the bold dotted line. The locus projection mimics the shape of the lower extent of FV flow, suggesting that near the boundary between the FV regime and the other two regimes, the lift coefficients in the three flow regimes do not strongly differ from one another.

3.5 Discussion: steady flow regimes

This section develops a few important ideas. First, it establishes that spanwise variation in the cavity length leads to re-entrant jet velocities that are well approximated by (3.2a ) and (3.2b ). Second, it verifies that the linearized slope of the cavity closure line ( $\bar{{\it\Phi}}$ ) is a suitable metric for predicting the presence of a destabilizing re-entrant jet. It should be recalled that the spanwise non-uniformities of the cavity are caused by combined 3D effects, gravity and nonlinear section forces, the gross effects of which are represented quite well by the nonlinear lifting-line model presented in § 1.2.2.

The division between the three flow regimes has been made on the basis of the stability of the flow. It should be noted that unsteadiness should not be confused with instability, as some stable flows (e.g. PV flow) may be inherently unsteady, even while locally or globally stable. A limitation applies to the flow-regime criteria developed in § 3.2. Callenaere et al. (Reference Callenaere, Franc, Michael and Riondet2001) and Franc & Michel (Reference Franc and Michel2006) point out that thin cavities (which occur at small attack angles) cause correspondingly thin re-entrant jets to develop. These jets are more severely affected by wall friction than are thicker jets; the thinness of the cavities means that the jets often collide with – and become re-entrained in – the external flow. Thus, for small angles of attack ( ${\it\alpha}\lesssim 5^{\circ }$ ), the thin jet is not sufficient to perturb the flow away from its locally stable configuration, so the $\bar{{\it\Phi}}=45^{\circ }$ condition is not expected to reflect the stability of the flow at such small attack angles. However, when small cavities with closure angles of $\bar{{\it\Phi}}\approx 45^{\circ }$ were subjected to external perturbations (air injections, small ripples, spray sheet, etc.), the flow often underwent a transition to PV flow, from which FV flow did not resume. At such small attack angles, $\bar{{\it\Phi}}=45^{\circ }$ is retained as the division between flow regimes – not because it necessarily signals instability caused by a re-entrant jet, but because it is a convenient and repeatable visual distinction which belies a tenuously stable flow configuration.

The establishment of three ventilated flow regimes is important for two reasons. First, it standardizes the definitions under which ventilated flow may be discussed and results disseminated. Previous studies have used disparate terminology. Second, it lays a groundwork from which to begin investigating the mechanisms of transition between ventilation regimes. Finally, a systematic method of characterizing the flow regimes is also necessary for establishing scaling relations for stability boundaries, more of which will be discussed in § 4.2.1.

The overlapping zones in figure 16 are of interest because they represent bi-stable conditions, where the flow may take on either one of the locally stable regimes. Factors governing the regime of a given flow include hysteretic effects with movement through the parametric map (such as during flow acceleration and deceleration) or external perturbations to the flow. Transition between two locally stable regimes is possible wherever the stability regions of those respective regimes abut or overlap one another. Overlaps along the $Fn_{h}$ -axis are particularly interesting, and merit further investigation. The stability boundaries were measured in accelerating and decelerating conditions (described further in § 4), so there is the possibility that some inertial and wake-memory effects are present. All efforts were made, however, to minimize such effects when populating the boundaries of figure 16 (see appendix B), so the overlaps are thought to be indicators of true bi-stable flow conditions, and not just transient effects.

4.1 Ventilation formation

Formation mechanisms promote the growth and stability of a cavity, causing the flow to transition from an FW or PV regime to a PV or FV regime respectively (see figure 17). Formation mechanisms can be further decomposed into two sequential stages, as follows.

Inception is the transition from FW flow to PV flow, which marks the first stage in the formation of a ventilated cavity. The first visual evidence of persistent air entrainment is sufficient to classify inception. Triangular symbols along the upper boundary of the FW region for ${\it\alpha}\geqslant 14^{\circ }$ in figure 16 indicate spontaneous ventilation inception (§ 4.1.1). Ventilation inception may also occur in zones 1 and 2 of figure 16, where the FW regime overlaps the FV and PV regimes respectively (described in § 4.1.2).

Stabilization is the transition from PV flow to FV flow, which completes ventilation formation. Stabilization is said to occur when the flow satisfies the criteria established for FV flow in § 3.2. Circles along the upper boundary of the PV region in figure 16 denote spontaneous stabilization events. Stabilization also immediately follows inception in zone 1 of figure 16. Under suitable conditions, stabilization can occur in zone 3 as well.

The two stages of ventilation formation can occur on multiple time scales and through different physical processes. Spontaneous formation mechanisms occur by self-initiated features of the flow. Perturbation-induced formation mechanisms, on the other hand, occur when some external influence perturbs the flow away from its locally stable state.

4.1.1 Spontaneous formation

Spontaneous formation mechanisms were examined by accelerating the towing carriage (constituting an upward vertical trajectory through the space of figure 16). Inception and stabilization were recorded to form the respective boundaries.

Figure 18. Spontaneous stall-induced ventilation formation occurring at ${\it\alpha}=15^{\circ }\approx {\it\alpha}_{s}$ , $AR_{h}=1$ . Here, $Fn_{h}$ is given at the instant of each photograph. Blue lines denote streamlines, green lines and hatching indicate entrained air, black hatching indicates boundary layer separation and red arrows indicate air-ingress paths. A movie of this process is also included in the online supplementary material (movie 1).

Stall-induced formation occurs when the angle of attack is set at or above the stall angle ( ${\it\alpha}\geqslant {\it\alpha}_{s}$ ). An example of the stall-induced ventilation formation process is depicted in figure 18, along with experimental photos of the hydrofoil suction surface. The flow begins in the FW regime. A large leading edge vortex creates a stagnant region of wetted separation, indicated by black hatching. In (a), the low pressures on the hydrofoil surface create a steep depression of the free surface, forcing proximity between the free surface (an air source) and the separated flow (ventilation-prone flow). At the same time, small vortical disturbances develop near the toe of the free-surface depression. Such vorticity development was noted in the waves generated by an unyawed surface-piercing hydrofoil by Pogozelski, Katz & Huang (Reference Pogozelski, Katz and Huang1997). The vortical structures at the free surface temporarily breach the surface seal, admitting air into the separated flow. If enough air is admitted, then it becomes stably entrained, constituting inception, shown in (b). After inception, the flow is classified as PV. As the flow velocity increases, the suction-side dynamic pressure negates the hydrostatic pressure to increasing depths and the sectional values of ${\it\sigma}_{c}$ decrease, causing the cavity to grow in both length and depth. As the cavity grows, it modifies the local chordwise pressure gradients, causing the separated flow to propagate ahead of the cavity (Tassin-Leger & Ceccio Reference Tassin-Leger and Ceccio1998), creating a moving front of ventilation-prone flow. The instant when the cavity meets the conditions for FV flow given in § 3.2 ( $D=h$ and $\bar{{\it\Phi}}<45^{\circ }$ ) is shown in (c). This constitutes stabilization, beyond which point the flow is classified as FV. Stall-induced formation is a quasi-steady process, wherein the flow takes on an evolving state of equilibrium as the speed, and hence the pressure distribution, changes. As a result, cavity formation takes place on a time scale governed by the acceleration of the hydrofoil.

Another case of spontaneous ventilation formation occurs when air enters the tip vortex at a downstream location and is transported upstream through the aerated vortex core. ‘Tip vortex inception’ was observed by Breslin & Skalak (Reference Breslin and Skalak1959) and Swales et al. (Reference Swales, Wright, McGregor and Rothblum1974) only at large angles of attack and small immersion depths (described in § 1.1). In the present experiments, air entered the tip vortex through the bubbly wake of the blunt trailing edge of the foil. A process sketch and experimental photos are shown in figure 19. The flow begins in the FW regime, with a base-vented cavity attached to the blunt trailing edge. Eddies in the wake draw air bubbles into the low-pressure core of the tip vortex, where they coalesce into an aerated vortex core, shown in (a). The coalesced bubble migrates up the low-pressure core towards the leading edge of the foil, where it encounters a favourable pressure gradient, as shown in (b). In (c), the air reaches the region of separated flow and expands rapidly upwards and towards the foil trailing edge. Fully ventilated flow is attained in a fraction of a second, with no appreciable dwell in the PV regime. Tip-vortex-induced formation occurred only at $AR_{h}=1.0$ , at yaw angles very near the stall boundary ( ${\it\alpha}\approx 15^{\circ }$ ), and at $Fn_{h}\approx 3.5$ .

Figure 19. Tip vortex ventilation formation occurring at angles near stall, ${\it\alpha}=15^{\circ }\approx {\it\alpha}_{s}$ , $AR_{h}=1$ . Here, $Fn_{h}$ is given at the instant of each photograph. Time is increasing from left to right at a constant Froude number for the case shown. The process may be seen in movie 2 of the online supplementary material.

Figure 20 depicts a surface-flow visualization at $Fn_{h}=2.5$ , ${\it\alpha}=14^{\circ }$ , $AR_{h}=1.0$ , corresponding to a location near the upper right corner of the FW stability region (in zone 1) in figure 16. A large separation bubble is clearly visible, indicated by the forward-swept paint streaks and black dashed outline. The separation bubble is conducive to both stall-induced formation and tip-vortex-induced formation. In the former case, air ingress occurs through disturbances in the thin layer of liquid between the separation bubble and the free surface, indicated by the free-surface profile and superimposed arrow in (a). In the latter case, air ingress occurs when a tip vortex develops of sufficient strength to transport air upstream to the separation bubble, shown in (b). Both stall-induced formation and tip-vortex-induced formation result when naturally occurring air paths reach the regions of flow separation.

Figure 20. Surface-flow visualization for ${\it\alpha}=14^{\circ }$ , $Fn_{h}=2.5$ , $AR_{h}=1.0$ , showing two possible mechanisms of ventilation inception (the first stage of ventilation formation). Red arrows indicate the respective ingress paths of air. (a) The free-surface profile immediately preceding spontaneous stall-induced inception. (b) The aerated tip vortex immediately preceding tip-vortex-induced inception.

The time histories of instantaneous lift, drag and moment coefficients of the two trials pictured in figures 18 and 19 are plotted against the instantaneous Froude number in figure 21. It should be noted that at small $Fn_{h}$ , the denominators of (3.3) vanish, and fluid inertial forces are large relative to the quasi-steady (velocity-dependent) load. As a result, $C_{L}$ , $C_{D}$ and $C_{M}$ may not be representative of steady-state results for $Fn_{h}<1$ . The grey and black traces describe two separate runs, each comprising an acceleration and deceleration cycle. Dashed arrows indicate the direction of the traces with increasing time. Symbols indicate steady-state force and moment coefficients. It should be noted that the steady-state data shown are projections of the same points shown in figures 12–15, plotted for a fixed ${\it\alpha}$ at different values of $Fn_{h}$ . The lettered markers indicate the corresponding panels in figures 18 and 19. The trace corresponding to stall-induced formation (solid black line in figure 21) follows the PV and FV markers during acceleration and deceleration, with the forces passing gradually through the inception and stabilization events at increasing values of $Fn_{h}$ . In the case of tip-vortex-induced formation (dashed grey line in figure 21), the loads pass through the FW points during the acceleration phase. At $Fn_{h}=3.5$ , inception and stabilization occur rapidly at a constant speed, indicated by a precipitous drop in $C_{L}$ and $C_{M}$ . During deceleration, the measured forces and moment follow the FV and PV points, overlapping with the deceleration section of the stall-induced formation case.

Figure 21. Time histories of lift, drag and moment coefficients for stall-induced (as depicted in figure 18) and tip-vortex-induced (as depicted in figure 19) ventilation formation at ${\it\alpha}=15^{\circ }$ , $AR_{h}=1$ . Dashed arrows indicate the acceleration and deceleration stages of the runs. Steady-state force/moment coefficients for all runs at ${\it\alpha}=15^{\circ }$ are plotted as open symbols, with measurement uncertainties and standard deviations indicated respectively by grey and black horizontal bars.

4.1.2 Perturbation-induced formation

In addition to the spontaneous formation processes observed at the boundaries of the stability regions, ventilation formation may be triggered by an external perturbation to an otherwise steady flow. When the parameters of a flow are inside one of the overlapping zones in figure 16, a suitable perturbation will destabilize the locally stable flow state and initiate a jump to a lower-energy state (FW to FV via inception and stabilization in zone 1, FW to PV via inception in zone 2, and PV to FV via stabilization in zone 3).

Figure 22 shows a sketch of a perturbation-induced ventilation formation process that would occur in zone 1. In (a), the flow is FW. Compared with the case of stall-induced formation, the leading edge separation bubble is small and the large Froude number means that the free surface is not so steeply deformed, so the free surface never approaches the boundary of flow separation. As a result, a thin layer of attached flow exists just below the free surface (indicated by a dashed line), sealing the ventilation-prone flow from the ingress of air. In (b), a high-pressure jet of air is injected at the junction of the leading edge and the free surface, introducing vorticity and bubbles into the flow at the free surface to break the surface seal. In (c), a small cavity is entrained and grows rapidly from the point of inception towards the immersed trailing edge until it reaches the lower-energy FV state. Entrainment and stabilization are nearly concurrent in this case, with little to no dwell in the PV regime. Fully ventilated flow can develop in approximately $1/10\text{th}$ of a second following perturbation.

Figure 22. Perturbed ventilation formation occurring at sub-stall, ${\it\alpha}=10^{\circ }<{\it\alpha}_{s}$ , $AR_{h}=1$ , $Fn_{h}=2.75$ (inside zone 1 of figure 16). Blue lines denote streamlines, green lines and hatching indicate entrained air, black hatching indicates boundary layer separation, the blue dashed line indicates the free-surface seal and red arrows indicate air-ingress paths. The process may be seen in movie 3 of the online supplementary material.

Figure 23 shows the surface-flow pattern at ${\it\alpha}=10^{\circ }$ , $Fn_{h}=2.5$ , $AR_{h}=1.0$ . The small yaw angle creates a short leading edge separation bubble, where flow reversal has swept several of the forwardmost columns of paint dots towards the leading edge. The free surface in the FW regime is superimposed as line (a). When air injection is initiated, the depression of the free surface permits air to enter the small bubble and expand, shown by line (b). Again, experimental observations support the hypothesis that inception occurs when an air supply is made available to ventilation-prone flow.

Figure 23. Surface-flow visualization for ${\it\alpha}=10^{\circ }$ , $Fn_{h}=2.5$ , $AR_{h}=1.0$ . The dashed line depicts the leading edge separation bubble caused by thin-airfoil stall. (a) The free-surface profile prior to ventilation inception and (b) the air-filled depression created by the air-jet perturbation.

Time histories of the lift, drag and moment coefficients of the process pictured in figure 22 are plotted against the instantaneous Froude number in figure 24. The resulting hysteresis loop is similar to that created by tip-vortex-induced formation in the sense that large changes in hydrodynamic loading occur on a short time scale, and at otherwise steady flow conditions.

Figure 24. Time histories of lift, drag and moment coefficients for perturbation-induced ventilation formation at ${\it\alpha}=10^{\circ }$ , $AR_{h}=1$ . The lettered markers correspond to the panels in figure 22. Dashed arrows indicate the acceleration and deceleration stages. Steady-state force/moment coefficients for all runs at ${\it\alpha}=10^{\circ }$ are plotted as open symbols, with measurement uncertainties and standard deviations indicated respectively by grey and black horizontal bars.

4.2 Ventilation elimination

Elimination encompasses processes that detract from the size or stability of a ventilated cavity. They describe the movement from an FV or PV regime towards a PV or FW regime respectively (see figure 17). As with formation mechanisms, elimination mechanisms can be broken down into two stages, given as follows.

Washout marks the transition from FV flow to PV flow. Washout is said to occur when the criteria for fully ventilated flow given in § 3.2 are no longer satisfied. This typically corresponds to a large-scale shedding of the cavity, after which the cavity topology fluctuates visibly. Squares along the lower edge of the FV region in figure 16 denote the washout event.

Rewetting marks the termination of air entrainment. It was observed that air was frequently ejected from the flow by laminar/turbulent reattachment of the flow at low speeds and angles of attack. It is said to occur when there ceases to be any visual evidence of air entrainment on the suction or pressure surfaces; there may still be bubbly flow entrained in the separated wake of the blunt trailing edge. Typically, the connection between the cavity and the free surface is terminated as the shallowly immersed sections experience flow reattachment. Isolated pockets of air cling to the leading edge of the hydrofoil until widespread reattachment of the flow sweeps the bubbles away. The ‘X’ symbols at the lower edge of the PV region in figure 16 indicate observed rewetting events.

Figure 25. Ventilation elimination process. Flow speed is decreasing from (a) to (c). Here, $Fn_{h}$ is given at the instant of each photograph. Blue lines denote streamlines, green lines and hatching indicate entrained air, and red arrows indicate air egress paths. The process may be seen in movie 4 of the online supplementary material.

Ventilation elimination was investigated by starting from an FV state and decelerating the hydrofoil, describing a downward trajectory in figure 16, during which washout and rewetting were recorded to form the stability boundaries. A sketch of ventilation elimination is shown in figure 25. The flow begins in an FV condition. In (a), a re-entrant jet develops on the deeply immersed hydrofoil section (shown by blue arrows). The cavity closure angle is shallow ( $\bar{{\it\Phi}}<45^{\circ }$ ), so the jet flow does not impinge on the upstream flow. As the hydrofoil decelerates, the cavity remains attached to the immersed tip, but reduces in its length, causing the angle of the cavity closure line to increase. In (b), the mean cavity closure angle just exceeds $45^{\circ }$ , imparting an upstream velocity component to the re-entrant jet, per (3.2). The jet pinches off a large portion of the cavity, causing it to be shed downstream (washout), leaving the flow in a PV state, from which FV flow does not resume. As the flow velocities decrease further in (c), the partial cavity is confined to a shrinking region near the leading edge of the foil and the free surface. Eventually, the remaining air is ejected by laminar/turbulent reattachment of the flow, constituting rewetting and the return to the FW regime.

The time history of lift, drag and yawing moment coefficients can be observed during the deceleration stages of figure 26. Following ventilation formation (which occurred via external perturbation for the case shown), the instantaneous forces and moments pass through the steady-state FV points, but return to the PV or FW regime at very small values of $Fn_{h}$ . The yawing moment coefficient most clearly shows the measured $C_{M}$ returning to the FW and/or PV values. It should be noted that when $Fn_{h}<0.5$ , the denominators of (3.3) are vanishingly small, so any noise in the signal is amplified. For this reason, the return of the hydrodynamic forces and moments to the FW values at low speeds is not obvious in the time histories shown. It should also be noted that in the neighbourhood of washout, indicated by marker (b), the hydrodynamic force/moment coefficients in the FW, PV and FV regimes are nearly coincident; this corroborates the observations from figure 15 that the intersection of $C_{L}$ values between the FW and FV regimes follows the lower boundary of the FV flow regime.

Figure 26. Time histories of lift, drag and moment coefficients for the ventilation elimination process at ${\it\alpha}=15^{\circ }$ , $AR_{h}=1$ . Dashed arrows indicate the acceleration and deceleration stages. The lettered markers correspond to the panels in figure 25. Perturbation-induced formation was used to achieve FV flow prior to deceleration. Steady-state force/moment coefficients for all runs at ${\it\alpha}=15^{\circ }$ are plotted as open symbols, with measurement uncertainties and standard deviations indicated respectively by grey and black horizontal bars.

4.2.1 Scaling of washout stage

The experimentally observed washout boundaries from the present experiment are plotted as open symbols in figure 27 as functions of the instantaneous lift coefficient $C_{L}$ and $Fn_{h}$ for the three immersed aspect ratios ( $AR_{h}=0.5,1,1.5$ ). The plotted data denote the combination of $Fn_{h}$ and $C_{L}$ at the moment that washout was visually observed, e.g. in panel (b) of figure 25 and marker (b) in figure 26. Data from Breslin & Skalak (Reference Breslin and Skalak1959) for hydrofoils with circular-arc and cambered NACA sections at four immersed aspect ratios ( $AR_{h}=0.5,1,1.5,2$ ) are plotted as filled symbols. The points indicate the boundary at which Breslin & Skalak (Reference Breslin and Skalak1959) observed the elimination of ventilated cavities; the specific stage of elimination was not specified in the original report, but is judged from context to be the cessation of FV flow, which has been defined as washout in the present work. Equation (1.2), shown as the black dashed line, represents the lower bound on stable FV flow proposed by Breslin & Skalak (Reference Breslin and Skalak1959).

Figure 27. Scaled washout boundary in $C_{L}$ , $Fn_{h}$ space. Data from the present experiments at three immersed aspect ratios, plotted as open symbols, and the experimentally observed boundary from Breslin & Skalak (Reference Breslin and Skalak1959) for cambered and circular-arc hydrofoils at four immersed aspect ratios, plotted as filled symbols, are shown. Equation (1.2) overpredicts the washout Froude numbers from the present experiments. The projected loci of lift-intersection points closely match the washout boundaries, suggesting that the lift coefficient is similar between the three flow regimes at washout.

It should be noted that the boundary given by (1.2) is valid only for $Fn_{h}\geqslant 3$ (Breslin Reference Breslin1958), above the horizontal line in figure 27. This restricts usage of the boundary to relatively low values of $C_{L}$ . It should be recalled also that Breslin & Skalak (Reference Breslin and Skalak1959) defined (1.2) such that $C_{L_{w}}$ is the instantaneous lift coefficient, measured when the hydrofoil is in an FW flow at the conditions under which ventilation elimination occurred. This makes (1.2) difficult to apply directly, because the lift coefficient may be unknown for FW flows at the combination of ${\it\alpha}$ and $Fn_{h}$ observed at the moment of washout. It should be recalled that in figure 15, the approximated intersection points between affine fits through the FV and FW lift coefficients formed a locus of intersection points. In order to justify the application of (1.2) to lift coefficients measured in the FV regime, the orthographic projections of these loci are plotted as bold lines in figure 27, where the lines closely follow the experimentally observed washout boundaries. This means that, at the conditions of washout, the lift coefficients in the FV, PV and FW regimes are approximately equal, so the measured $C_{L}$ at washout is a suitable approximation of $C_{L_{w}}$ . As shown in figure 27, equation (1.2) overpredicts the washout $Fn_{h}$ for a given $C_{L}$ . In fact, it passes through only a few of the experimental points from Breslin & Skalak (Reference Breslin and Skalak1959), while washout was never observed at $Fn_{h}>3$ in the present experiments.

There are several reasons why the experimentally observed washout boundaries would differ between the present study and that of Breslin & Skalak (Reference Breslin and Skalak1959). The first is the difference in section profiles. As mentioned in § 4.1.3, the present model has a rectangular section aft of mid-chord, while both of the models used by Breslin & Skalak (Reference Breslin and Skalak1959) had sharp trailing edges and convex curvature on the suction surfaces aft of mid-chord. As a result, washout may have occurred at higher Froude numbers for the latter model because the re-entrant jet was enhanced by an adverse pressure gradient developed on the wetted portion of the suction surface. Another factor is the difference in size between the two models. The present model is approximately 4–5 times as large as those used by Breslin & Skalak; the smaller model would be more susceptible to surface-tension effects (Wetzel Reference Wetzel1957), where spray sheets close and cut off the supply of air to the cavity in a process called cavity ‘choking’ (Elata Reference Elata1967). Equation (1.2) also overpredicts the washout boundary of Breslin & Skalak (Reference Breslin and Skalak1959) for $3<Fn_{h}<6$ . This occurs because (1.2) was derived from a linearized fit at very large $Fn_{h}$ and vanishingly small $C_{L}$ . Equation (1.2) does form an upper bound on all other presented experimental data, so it is appropriate to consider it as a boundary above which FV flow is likely to be stable, but not a boundary below which full ventilation is necessarily unstable. To develop an improved washout scaling relation, the kinematic stability boundary on the cavity shape ( $\bar{{\it\Phi}}=45^{\circ }$ ) can be used, along with models for the cavity length ( $L_{c}$ ), lift coefficient ( $C_{L}$ ), Froude number ( $Fn_{h}$ ), aspect ratio ( $AR_{h}$ ) and angle of attack ( ${\it\alpha}$ ). An initial attempt was presented by Harwood et al. (Reference Harwood, Brucker, Miguel, Young and Ceccio2014), but an improved scaling approach was deemed necessary.

Figure 28 schematically depicts an FV cavity. A sectional cut is taken at a dimensionless depth of ${\it\kappa}=z^{\prime }/h$ and a linear approximation of the cavity closure angle is made between this section and the immersed tip – a distance of $(1-{\it\kappa})h$ . The stability limit of $\bar{{\it\Phi}}=45^{\circ }$ is equivalent to a unity-slope condition on this linearization. To model the cavity length at the section in question, equation (1.8) is used, giving

(4.1) $$\begin{eqnarray}\frac{L_{c}(z^{\prime }={\it\kappa}h)}{c}=(1-{\it\kappa})\frac{h}{c}=4.62\frac{{\it\alpha}_{2D}(z^{\prime }={\it\kappa}h)}{{\it\sigma}_{c}(z^{\prime }={\it\kappa}h)}.\end{eqnarray}$$

Figure 28. Sketch of the cavity topology. The cavity stability criterion of § 3.2 is expressed using a two-point linearization of the cavity closure line. When the slope of the linearized cavity closure line approaches unity ( $\bar{{\it\Phi}}=45^{\circ }$ ), it is hypothesized that washout is imminent.

The next step is to infer the sectional lift coefficient at the representative section from the 3D lift coefficient, which requires a priori knowledge of the distribution of lift along the span. A model such as the lifting-line model presented in § 1.2.2 could be used, but an analytical approximation is desirable in this case, in order to derive a closed-form scaling relation. An elliptical shape function may be applied to the lift distribution as a simple approximation for rectangular wings (Glauert Reference Glauert1943), and one that retains the salient features of the lift distribution shown in figure 3. The low aspect ratio is not a concern because the elliptical distribution can be scaled such that integration along the span yields the experimentally measured 3D lift coefficient. Thus, the sectional lift coefficient at the representative section is taken to be

(4.2) $$\begin{eqnarray}C_{l_{2D}}(z^{\prime }={\it\kappa}h)=\frac{4}{{\rm\pi}}C_{L}\sqrt{1-(2{\it\kappa}-1)^{2}}.\end{eqnarray}$$

The effective angle of attack can be found by substituting $L_{c}/c=(1-{\it\kappa})AR_{h}$ into (1.9) to find $a_{0}$ , giving

(4.3) $$\begin{eqnarray}\displaystyle & & \displaystyle {\it\alpha}_{2D}(z^{\prime }={\it\kappa}h)=\arcsin \left(\frac{C_{l_{2D}}(z^{\prime }={\it\kappa}h)}{a_{0}(z^{\prime }={\it\kappa}h)}\right)\approx \frac{C_{l_{2D}}(z^{\prime }={\it\kappa}h)}{a_{0}(z^{\prime }={\it\kappa}h)}\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{4C_{L}\sqrt{{\it\kappa}-{\it\kappa}^{2}}\{4{\rm\pi}[(1-{\it\kappa})AR_{h}]^{3}-4{\rm\pi}[(1-{\it\kappa})AR_{h}]^{2}+3{\rm\pi}[(1-{\it\kappa})AR_{h}]+2\}}{{\rm\pi}\{{\rm\pi}^{2}[(1-{\it\kappa})AR_{h}]^{3}-4{\rm\pi}[(1-{\it\kappa})AR_{h}]^{2}+9{\rm\pi}(1-{\it\kappa})AR_{h}+2{\rm\pi}\}}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

It follows from (1.12) that

(4.4) $$\begin{eqnarray}{\it\sigma}_{c}=\frac{2}{Fn_{h}^{2}}\frac{z^{\prime }}{h}=\frac{2}{Fn_{h}^{2}}{\it\kappa},\end{eqnarray}$$

which may be substituted into (4.1), along with (4.3). The representative section of the hydrofoil is taken at mid-span ( ${\it\kappa}=0.5$ ), with the reasoning that any re-entrant jet posing a threat of destabilization to the flow is anticipated to be reflected about the deeply submerged portions of the cavity. Solving for $Fn_{h}$ yields the semi-theoretical washout Froude number,

(4.5) $$\begin{eqnarray}Fn_{h}=\frac{{\rm\pi}}{4}\sqrt{\frac{{\displaystyle \frac{{\rm\pi}}{2}}AR_{h}^{4}-4AR_{h}^{3}+18AR_{h}^{2}+8AR_{h}}{2.31C_{L}({\rm\pi}AR_{h}^{3}-2{\rm\pi}AR_{h}^{2}+3{\rm\pi}AR_{h}+4)}}.\end{eqnarray}$$

The boundaries given by (4.5) are shown in figure 29, again with the data of Breslin & Skalak (Reference Breslin and Skalak1959) and (1.2). The present experimental washout boundary is captured very well by (4.5). The data of Breslin & Skalak (Reference Breslin and Skalak1959) are reasonably well approximated by the present theory for $Fn_{h}\leqslant 3$ , while (1.2) better approximates the points at very small values of $C_{L}$ and $Fn_{h}>3$ .

Figure 29. Scaled washout boundary in $C_{L}-Fn_{h}$ space. Data from the present experiments and from Breslin & Skalak (Reference Breslin and Skalak1959) are plotted. The stability boundary of Breslin & Skalak (Reference Breslin and Skalak1959) (1.2) overpredicts the washout Froude numbers, while the present semi-theoretical boundary (4.5) captures the present experimental data across the range of $C_{L}$ , and captures the data of Breslin & Skalak (Reference Breslin and Skalak1959) for $C_{L}\gtrsim 0.6$ .

The value of $C_{L}$ is often not known a priori, requiring it to be inferred as well. Again, an approach such as the lifting-line model from § 1.2 may be used to predict $C_{L}$ for given operating conditions. However, it is more convenient to implement an analytical model of $C_{L}$ as function of ${\it\alpha}$ , $AR_{h}$ and $Fn_{h}$ , so that the washout boundary may be represented as a function of ${\it\alpha}$ and $Fn_{h}$ without the necessity of solving (1.19) for each condition. In § 4, it was shown that the instantaneous hydrodynamic load coefficients in unsteady flows did not differ materially from those measured in steady flows. A model that suitably predicts the steady-state lift coefficient at low $Fn_{h}$ may, as a result, be substituted into the $C_{L}$ term in (4.5). It should be recalled that (1.20) may be used to model $C_{L}$ by assuming a representative value of $a_{0}$ . Equation (1.22) can be used again to calculate the lift-weighted mean, $a_{0}^{\ast }$ , but doing so requires that the cavity length and lift distributions are known. The distribution of cavity lengths is constrained by the assumption of $\bar{{\it\Phi}}=45^{\circ }$ to be

(4.6) $$\begin{eqnarray}\frac{L_{c}({\it\kappa})}{c}=AR_{h}(1-{\it\kappa}).\end{eqnarray}$$

Equation (4.6) may be substituted into (1.9) to obtain $a_{0}({\it\kappa})$ . The assumed elliptical lift distribution is scaled to yield an integral of one, giving the shape function,

(4.7) $$\begin{eqnarray}E({\it\kappa})=\frac{4}{{\rm\pi}}\sqrt{1-(2{\it\kappa}-1)^{2}}.\end{eqnarray}$$

By weighting $a_{0}({\it\kappa})$ with $E({\it\kappa})$ , equation (1.22) becomes, under a change of variables,

(4.8) $$\begin{eqnarray}a_{0}^{\ast }=\int _{0}^{1}a_{0}({\it\kappa})E({\it\kappa})\,\text{d}{\it\kappa},\end{eqnarray}$$

which is evaluated using numerical integration. The lift-weighted mean value $a_{0}^{\ast }$ is substituted into (1.20), and the predicted $C_{L}$ is in turn substituted into (4.5). Doing so yields a function of $Fn_{h}$ , ${\it\alpha}$ and $AR_{h}$ . As before, solving for $Fn_{h}$ results in a washout Froude number, plotted in figure 30 as a boundary in ${\it\alpha}$ – $Fn_{h}$ space. A value of ${\it\kappa}=0.5$ is again assumed. The boundaries from the present experiments at three aspect ratios ( $AR_{h}=0.5,1,1.5$ ) are shown as open symbols. Experimental points from Breslin & Skalak (Reference Breslin and Skalak1959) (circular-arc hydrofoil at $AR_{h}=0.5$ , 1, 1.5 and 2) and Swales et al. (Reference Swales, Wright, McGregor and Rothblum1974) (biogival hydrofoil at $AR_{h}=2$ ) are shown for comparison. The two sets of data shown for Swales et al. (Reference Swales, Wright, McGregor and Rothblum1974) are from tests conducted in a towing tank at atmospheric pressure and tests in a variable-pressure water tunnel under reduced ambient pressure. The experimental data represent a very diverse range of operating conditions, model sizes and model geometries (see table 1), which are in good general agreement with the semi-theoretical washout boundary.

Figure 30. Scaled washout boundary in ${\it\alpha}$ – $Fn_{h}$ space. Data from the present experiments and previous experiments are plotted as symbols. B & S, Breslin & Skalak (Reference Breslin and Skalak1959); S & W, Swales et al. (Reference Swales, Wright, McGregor and Rothblum1974) experiments in a variable-pressure water tunnel; 1, tests conducted at atmospheric pressure; 2, tests conducted at a reduced pressure of 4.67 kPa.