On the Collapse Structure of an Attached Cavity on a Three-Dimensional Hydrofoil

Fully developed sheet cavitation on ship propellers is a major cause of noise, vibration, and erosion. Although the final evaluation of a propeller design is based on model experiments, calculations of cavitation are becoming increasingly more important. Potential flow solvers are now the industry standards (e.g., Young and Kinnas (1)), but in the past decades, an increase in both Euler (e.g., Choi and Kinnas (2), and Schnerr et al. (3)) and Reynolds-averaged Navier-Stokes (RANS) (e.g., Kunz et al. (4)) codes is observed. However, up until now, these simulations are not able to capture the pressures radiated by cavitation or to predict erosion location and severity on propellers (ITTC (5)). To improve the description of the cavity behavior and especially the unsteady shedding in the form of cloud cavitation and in support of the rapidly expanding field of numerical simulation, this experimental research was started with a threefold goal: First, analyze the physical mechanisms of the instability of the cavity; second, build a data set of simple cavitating flows to be used as benchmark material for computational fluid dynamics validation; and third, extend the insights gained to guidelines for propeller design. Here, we focus on the description of the flow field around an attached cavity and its shedding mechanism.

Cavitation has been extensively tested in the past on two-dimensional hydrofoils (i.e., Franc and Michel (6)). However, cavitation on ship propellers is distinctly three dimensional due to the propeller’s three-dimensional geometry, the radially increasing velocity and change in blade loading, and a periodic change of inflow conditions due to the wake behind a ship’s hull. As studying cavitation on a rotating object is inherently more difficult, three-dimensional hydrofoils were designed with a spanwise variation in loading, resulting in a cavitation topology closely related to propellers. This allows for observations of the influence of controlled three-dimensional effects of the attached cavity. For this study, a span-symmetric three-dimensional hydrofoil is chosen, creating an isolated sheet cavity around the plane of symmetry. The hydrofoil is lightly loaded at the tunnel walls to avoid any interaction of the cavity with the tunnel boundary layer.

Crimi (7) studied the effect of sweep (skew) and concluded that the inception velocity increased with an increase in the skew angle. Hart et al. (8) investigated an oscillating three-dimensional finite-span hydrofoil and concluded that the cavity collapse was most violent when the oscillating frequency coincided with the natural shedding frequency of the cavity. de Lange and de Bruin (9) concluded that the reentrant of the two-dimensional hydrofoil was directed upstream, but in the three-dimensional case, the reentrant jet component normal to the closure line was reflected inward. As the pressure gradient is perpendicular to the closure line, the flow is deflected perpendicularly to the cavity closure line. Laberteaux and Ceccio (10) studied a series of swept wedges. The cavity plan form was significantly changed and the reentrant jet was directed into the cavity, allowing for a steady sheet that only shed cloud cavitation at the far downstream edge. Dang and Kuiper (11) numerically studied the reentrant jet on a hydrofoil with a spanwise varying angle of attack and found the reentrant jet direction to be strongly influenced by the cavity topology. The change in cavity shape was determined by loading and not by the sweep angle.

In this paper, high-speed recordings are presented with our interpretation for the shedding behavior for two distinct cases, a cavity of roughly half the chord length and a supercavity. The shedding mechanism for both cases differed from two-dimensional shedding—where the reentrant jet reaches the leading edge—but was governed by the three-dimensional topology of both hydrofoil and the attached cavity. A brief description of the setup of the experiment is given in Sec. 2. The observations are described and interpreted in Sec. 3 and conclusions are in Sec. 4.

The test object is a three-dimensional hydrofoil, previously used by Dang (12), with a chord length of $C=150mm$⁠, a span of $S=300mm$ (spanning the entire test section), and a spanwise varying angle of attack (Fig.3). This geometric angle of attack varies as sketched in Fig.4, rotating the sectional profile (NACA0009) around the midchord position. (See Ref. 13 for specific details.) Calculations by Koop et al. (14) indicate that the change in effective angle of attack is only $2deg$⁠. The hydrofoil was manufactured in both anodized aluminum and perspex. The aluminum hydrofoil was mounted with its suction side downward so that it can be easily filmed from below. The perspex hydrofoil was mounted with the suction side upward and was filmed from the pressure side so that the cavity could be viewed from inside of the cavity.

The boundary layer near the minimum pressure region will remain laminar at low Reynolds numbers and no cavity sheet will appear. When the boundary layer separates and a separation region is formed, a smooth and glassy cavity can appear. With the limited Reynolds numbers typically present at small scales, transition to turbulence does not occur unless the boundary layer is locally disturbed. When it does occur, natural transition to turbulence can temporarily suppress leading edge flow detachment (6). As a three-dimensional hydrofoil is used at moderate velocities, the sides of the attached cavity can be locally suppressed as the flow remains laminar, so roughness elements of $120μm$ were applied at the leading edge (4% chord length) as a turbulence tripping mechanism. The roughness elements can lead to local streaks of cavitation appearing next to the main cavity. At too low speeds, the entire detachment region near the leading edge may resemble an agglomeration of such streaks, which was observed at $5m∕s$⁠. The gas content was measured to be less than 0.1%, but the roughness will supply the degassed flow with ample nuclei for sheet cavitation to develop (15); incipient cavitation on roughness elements is typically observed when $σ$ equals the minimum pressure coefficient (16) and the nuclei content of the flow is no longer critical.

The camera used for the high-speed imaging is a Photron Ultima APX with a $10bit$ dynamic range, $1megapixel$ resolution at $2kHz$ with a maximum acquisition frequency of $120kHz$ (0.4% full resolution) with $2.6Gbyte$ memory. The lens is a Nikon AF Nikkor $50mm$⁠, used with a f-stop of 2.8. A New Wave Pegasus dual-head, high repetition, diode pumped Nd:YLF laser was used as a stroboscope, with a $180ns$ duration with a $10mJ$/pulse power at $1kHz$⁠.

The shedding process of the attached cavity is classified into three regimes. At high cavitation numbers $(σ>1.1)$⁠, the attached cavity was short in length and present over a wide part of the leading edge and hence mainly two dimensional. This cavity was shedding vortices intermittently; no large cloudy structures were identified. Such a closure was termed “open” by Laberteaux and Ceccio (10). At moderate cavitation numbers $(0.65>σ>1.1)$⁠, large structures were shed regularly. This intermediate regime was dominated by the three dimensionality of the cavity. Lowering the pressure further $(σ<0.65)$ created an attached cavity reaching a length comparable to the chord length of the hydrofoil. Shedding was then intermittent and irregular. The cavity spanned the entire foil and was once more mainly two dimensional.

Figure6 presents an example of an intermittently shedding (super) cavity at $σ=0.49$⁠, showing the shedding in detail. In Figs. 6.1–6.6, the location of the front of the reentrant jet is given by the arrow at the center. Although difficult to identify on photographs, on the recordings, it is clearly seen to move slowly forward. The breakup of the sheet in Figs. 6.1–6.10 started from the end into distinct vortices, moving upstream. These vortices later collapsed. This upstream movement of the cavity closure was only observed when the cavity length exceeded the chord length of the hydrofoil.

The shed flow structure in Figs. 6.7 and 6.8 consisted of primary spanwise and secondary streamwise vortices, similar to the turbulent shear flow structure observed behind steps and other mixing layers (18). Figure7 shows the formation of a spanwise vortex from Figs. 6.5–6.7 (showing intermediate images as well). Such a spanwise vortex system can be a result of a Kelvin–Helmholtz instability with a street of vortices with a positive strength (where the vorticity has the same sign as the hydrofoil’s circulation). A close-up of Figs. 6.6–6.9 is given in Fig.8, including all intermediate images. Bernal and Rosko (19) describe a structure that greatly resembles the presented shedding structure of spanwise and streamwise vortices, describing the structure of a helium-nitrogen mixing layer. The streamwise vortices originated as a single spanwise vortex warped around the primary spanwise vortices. The smaller scale vortices can be seen to be stretched around the periphery of the spanwise structures with an increase in their vaporous cores. Cavitation inception is first observed in these streamwise vortices in shear layers (18), but in the case of a sheet cavity, break-up vapor is trapped in the initial formation.

From Figs. 6.4–6.8 is visible that the front of the mixing moved forward and the breakup was cascading toward the leading edge. The breakup of the sheet started out as concave but the front drew parallel to the span as it progressed upstream. The front of the disturbance accelerated at a constant rate up to the mean stream velocity when reaching the leading edge, as determined from frame-by-frame analysis. The approximated location of the front at the center plane was identified and plotted in Fig.9.

The reentrant jet momentum depends on the pressure gradient in the closure region (20). The increase in collapse speed may be explained as follows. At the start of the collapse cycle, the cavity is a well-defined spatial structure with a convex closure. Due to the three-dimensional geometry with a symmetry plane, a stagnation point is present in the closure region only (Fig. 10.1). After the first pinch-off, the closure region of the cavity has changed from a convex into a concave or straight shape and the reattachment region has widened (Fig. 10.2) and widening further with each pinch-off (Fig. 10.3) as the cavity loses its three dimensionality. From observations at higher values of $σ$—presented below—it is observed that on a three-dimensional cavity, the reentrant jet diverges radially from the closure into the sheet when the cavity is fully grown.

In Fig.11, a full shedding cycle at $5m∕s$ and $σ=0.66$ of a regularly shedding cavity is shown, with the flow from top to bottom. The shedding was repeatable, constant in its shedding frequency, and always followed the same macrostructural collapse.

The shedding cycle of the cavity in Fig. 11 is divided into four phases: destabilization, primary and secondary shedding, followed by growth into its initial condition. There is a short overlap between primary and secondary shedding (and growth). The primary shedding is located at the midplane of the hydrofoil; the secondary shedding is visible at the sides of the region of the primary shedding as two distinct smaller vortices.

Phase 1 11.1–11.4 Initial disturbance

Phase 2 11.5–11.12 Primary shedding (cavity center)

Phase 3 11.9–11.16 Secondary shedding (cavity sides)

Phase 4 11.17–11.20 Growth

Figures 11.1–11.4 show the convex cavity, here considered fully grown. The lower part of the cavity interface was turbulent, while the cavity at the sides and near the leading edge was glassy and transparent. It is in the closure region where the cavity became turbulent first, not near the leading edge, as is typical of large structure shedding on two-dimensional hydrofoils. The reasons are twofold.

• 1

  The closure region in a two-dimensional flow would normally be followed by a stagnation line (parallel to the leading edge); here, it was a stagnation point at the midplane implying that the local pressure gradient was weakened. As indicated above, the momentum in the reentrant jet depends on the pressure gradient in the closure region, so the three-dimensional topology of the attached cavity resulted in a reentrant jet diverging radially into the cavity from its closure at the midspan position. Therefore, its forward momentum is diminished as it progressed into the cavity.

• 2

  At the sides of the sheet, the pressure gradient forces the flow over the sheet into the cavity roughly “mirroring” the streamlines at the interface contour, as sketched in Fig.12,12. de Lange and de Bruijn concluded that the reentrant jet of the two-dimensional hydrofoil was directly upstream, but in the three-dimensional case, the reentrant jet component normal to the closure line was reflected inward. As the pressure gradient was perpendicular to the closure line, the component tangential to the closure line remains unchanged. At the sides of the cavity, the reentrant flow had a very small spanwise component and was directed downstream. The spanwise component was largest when the cavity closure contour was at about $45deg$ with the incoming flow where the velocity component in the downstream direction of the reentrant flow was zero. When the sheet cavity was growing, flow from the sides was not obstructed, nor was it directed at the leading edge.

To distinguish between various directions of the reentrant flow, the term side-entrant jet is introduced. This term refers to that part of the reentrant flow that has a strong spanwise velocity component. The term reentrant jet is reserved for the reentrant flow that has a velocity component that is mainly streamwise. The reentrant flow is thus split up in reentrant and side-entrant jet components, even though at certain points of the flow, both terms may apply. Note that the side-entrant jet component, in contrast to the reentrant jet component, is not necessarily directed upstream. The term side entrant is introduced to emphasize the three-dimensional character of the flow. For the case presented, the side-entrant jets from both sides were flowing into the closure region of the sheet where they collided. Side-entrant jets of the reentrant flow do not reach the leading edge but may form an equally important source for the shedding.

Any fluid ejected upward through the cavity interface created a significant disturbance, isolating a small portion of vapor and creating a bubbly flow consisting of jet-entrained vapor. The velocity of a streamline at the cavity surface is measured at $VV=V01+σ$ (13). Although the velocity of the reentrant flow is difficult to measure, the velocity of the jets is unlikely to be an order of magnitude lower. Also, if we assume that during the shedding cycle the two side-entrant jets were converging for about a third of the shedding cycle $(15Hz)$⁠, the amount of fluid through a square millimeter—taking a homogeneous velocity distribution—at this velocity of $6.4m∕s$ was about $285mm3$ per $mm2$ cross section of the reentrant flow. At this rate, the cavity closure could be collecting fluid quickly even if the jets were thin.

The primary shedding originated at the collision region in the center of the sheet, see Figs. 11.5–111.12. However, only a portion was broken off from the main sheet and advected with the flow. Most of the cavity remained attached. This structure could be seen to roll up quickly in Figs. 11.5–11.8 by self-induction into a hairpin vortex. This structure grew significantly in height, on the order of the cavity length. The cavity closure after the cutoff of the hairpin vortex was temporarily turbulent—shedding a large cloudy structure—but reattached smoothly shortly thereafter. In order to visualize the reentrant flow more clearly, a series of additional images of the transparent hydrofoil is presented in Figs.14,15. The cavitation was filmed through the pressure side of the hydrofoil showing the internal structure of the cavity. The radially diverging reentrant flow is clearly visible in Fig. 14 (denoted as A) as waves on the jet surface reflected the laser light.

The reentrant flow directed upstream in a two-dimensional situation would be constrained in its lateral movement. The vapor interface at the leading edge was not visibly disturbed upon contact with this reentrant flow; its apparently low momentum did not lead to immediate shedding. As the side-entrant jets were aimed at the closure, it was here that the fluid first impinged on the interface. Therefore, the main cause for the detachment of the main structure was the side-entrant jet and not the reentrant jet.

The remaining topology of the sheet closure line in Figs. 11.5–11.12 was concave. The locally convex regions of the cavity are seen to shed a series of larger vortices, followed by a turbulent flow region. From observation, the secondary shedding greatly resembled the primary shedding. The secondary shedding disappeared when the closure was no longer concave.

The reentrant flow direction in the center was still directed radially outward. The main side-entrant jets and radially diverging reentrant jet were now converging in both downstream lobes of the remaining cavity shape (Figs.16,17). The secondary shedding was caused by the collision of these two flows. Basically, the main shedding as visible in Figs. 11.5–11.12 was repeated at both sides of the center plane, as visible in Figs. 11.9–11.18.

After the secondary shedding, the remaining cavity had a near-convex shape with two concave regions, denoted H in Fig.18, corresponding to Fig. 11.18. From these regions, the reentrant flow entered the cavity sideways, similar to the reentrant flow in Fig. 17 after the primary shedding. The reentrant flow from the closure of the cavity at the midplane and from the sides on the cavity—denoted B and C, respectively, in Fig. 18—remained present and collided with the side-entrant jets from H. Figure 15 shows this situation on the transparent hydrofoil. The movement of the front of the side-entrant flow (A) from these regions at Fig. 18-H can be seen, as the reentrant flow forced into the cavity collided with the reentrant flow from the plane of symmetry (B) and a frothy turbulent region was created upon impact at the lower corners. At the outer side, a continuous mixing is observed as the reentrant flow from the main flanks of the cavity (Fig. 18.C) continued to collide with the reentrant flow from (H). No large-scale shedding was observed at this point of the shedding cycle, as with each subsequent shedding, the scale and hence total jet momentum decreased, while the inflow and its momentum from the sides of the cavity remained constant. Without any further major disturbances allowing for a sheet topology change in the closure region, the cavity grew back into its original convex shape and side-entrant jets at the center plane collided once again repeating the process. The cavity did not reach a constant length.

The shedding of the sheet cavity of the three dimensionality is similar to a two-dimensional shedding, having its origin a disturbance of its interface, except that the disturbance occurs at the aft part of the sheet. The fluid impinging on the interface isolates a region of vapor, as sketched in Fig.19. If the interface is considered a streamline with a tangent velocity $VV=V01+σ$⁠, it is immediately apparent from contour integral of that velocity over the boundary of $S$ that circulation is detached and adverted with the flow. The impingement and detachment of this vapor structure are inertial in nature. The mixing layer with its region of high shear and strong vortices visibly generates vorticity.

From the experiment, investigations with the three-dimensional cavities follow that reentrant flow from the sides dictates the behavior of the shedding cycle. The reentrant flow from the sides depends on the cavity shape. Thus, the cavity topology largely dictates the reentrant flow direction.

The convex cavity planform leads to converging reentrant flow and that flow convergence leads to shedding in the cavity closure region. Reentrant flow was observed to reach the leading edge, which did not result in shedding. The reentrant flow can be moving both upstream and in the spanwise direction. The spanwise component of the reentrant flow is denoted as the side-entrant jet. For any convex cavity shape, the side-entrant components of the reentrant jet converge in the closure region of the sheet, creating a disturbance that causes local breakoff of the aft part of the main sheet structure. This converging of the side-entrant flow is suggested as a second shedding mechanism for attached sheet cavitation, in addition to the well-known mechanisms of the reentrant flow impinging on the cavity interface near or on sheet cavity detachment point. The cause of the shedding is the same: impingement of a high-momentum flow on the surface of the hydrofoil on the cavity interface. The attached cavity on the suction side of a propeller is always three dimensional. The isolated cavity as presented on the current hydrofoil—that is, not connected to the tip of the propeller—is present on some propeller designs and the isolated cavity is shown to be inherently unstable.

With a convex cavity closure line, side-entrant jets converge in the cavity closure region leading to a pinch-off of the aft part of the cavity. The observed (cavitating) vortices in the wake of the remainder of the attached cavity are similar to the spanwise and streamwise vortices in a planar mixing layer. It is concluded that the wake of an attached cavity shedding small-scale vortices is, in fact, a mixing layer with its characteristic wake structure.

The alternating shedding seen on the three-dimensional hydrofoil results in a distinct cycle. However, the two-dimensional or rotational-symmetric hydrofoil lacks the spanwise variation in pressure distribution in the center, resulting in the seemingly random local shedding along its cavity closure. Any disturbance at its closure will redirect the reentrant flow into converging side-entrant flow resulting in local shedding. The two-dimensional cavity has a highly three-dimensional structure, making it more difficult to study, either numerical or experimental, with the reentrant flow constantly changing direction and continuously converging in other locations along the span. The three-dimensional cavity is shown to have a repeatable collapse making it a more proper candidate for numerical validation studies of cloud shedding.

This research is funded by the Dutch Technology Foundation STW Project No. TSF.6170 and the Royal Netherlands Navy. See www.stw.nl for more details.