Suppressing tip vortex cavitation by winglets

Abstract

Despite the numerous remedies prescribed so far, tip vortex cavitation (TVC) remains a major issue in design and operation of diverse applications. In this paper, we experimentally investigate the effectiveness of winglets in suppressing TVC. An elliptical hydrofoil is selected as the baseline geometry and various winglets are realized by bending the last 5 or 10% of the span at ± 45° and ± 90° dihedral angles. To better focus on the physics of the problem, we have intentionally avoided any optimization on the geometries and our winglets are only smooth non-planar extensions of the original cross-section. Modifying no more than 3.7% of the lifting surface, lift-and-drag force measurements demonstrate that the hydrodynamic performances of the winglet-equipped hydrofoils are not substantially different from the baseline. Nevertheless, cavitation inception–desinence tests reveal that undeniable advantages are achieved by the winglets in TVC alleviation. It is found that the 10%-bent 90° winglets are more effective than the 45° cases, with − 90° (bent down toward the pressure side) performing superior to + 90°. For instance, the 90°-bent-downward winglet reduces the TVC inception index from 2.5 for the baseline down to 0.8 (a reduction of 68%) at 15 m/s freestream velocity and 14° incidence angle. In addition, the study on the bending length effect conducted for the 90° configurations shows that the 5%-bent winglets are not as striking as the 10% ones. Employing Stereo-PIV technique, the influence of winglets on non-cavitating flow structures is examined. For the most effective winglet (10%-bent 90°-downward), we observe that the maximum tangential velocity of the tip vortex falls to almost half of the baseline and the vortex core size increases significantly (by almost 70%). These effects are accompanied by a tangible reduction in the axial velocity at the vortex core leading to further mitigation of TVC.

Graphic abstract

1 Introduction

Occurrence of cavitation within tip trailing vortices is a challenging issue for axial hydraulic machines. For instance, in Kaplan turbines, tip vortex cavitation (TVC) is the dominant cavitation regime associated with a high risk of erosion (Roussopoulos and Monkewitz 2000; Escaler et al. 2006; Luo et al. 2016). Besides, TVC is reportedly known as the first type of cavitation that incepts in marine propellers, the occurrence of which may be accompanied by severe structural vibrations and noise emissions (Arndt 2002; Shen et al. 2009; Arndt et al. 2015; Song et al. 2017; Lee et al. 2018a). Souders and Platzer (1981) tested four TVC mitigation techniques including adding artificial roughness to the tip, bulbous tip, and passive or active water injections into the vortex core and evaluated all of them effective with minimal decreases in lift generation. The injection-based methods have been investigated by many researchers over years. For instance, Fruman and Aflalo (1989) employ drag-reducing polymer solutions and show that the ejection of semi-dilute solutions significantly delays the TVC inception with no considerable changes in the hydrodynamic performance of the blade. Chahine et al. (1993) highlight the importance of viscoelastic properties of the injected solution and also the ejection location on the propeller blades. Another similar study (Chang et al. 2011) reveals that low fluxes of mass injection reduce flow unsteadiness at the vortex core while higher injection rates result in a moderate increase in the viscous core radius. Water injection is also reported to noticeably decrease the induced noise levels of TVC (Lee et al. 2018a). More recently, Park et al. (2014) demonstrated that attaching a flexible thread to the blade tips of a propeller could effectively attenuate TVC. This method was then adopted and tested for fixed elliptical hydrofoils by Lee et al. (2018b).

Wingtip vortices are a source of concern in aeronautic applications, too. The ‘vortex drag’ or ‘lift-induced drag’ may comprise up to 40% of the total cruise drag of commercial airplanes with significant economic and environmental impacts due to the added fuel consumption (Kroo 2005). These vortices may cause hazardous effects on flight safety as well, which obliges the air traffic control to increase the distances between following aircrafts at the expense of increased costs and delays (Spalart 1998). A common remedy for abating tip vortices is appending winglets to airplane wingtips. In 1897, Lanchester patented the idea of attaching a vertical endplate to planar wings for induced drag reduction (Whitcomb 1976). This configuration was rejected in practice because of the overall increased drag due to the enlarged wetted area and related viscous drag (McLean 2005). Whitcomb was probably the first to ascertain that this drawback could be solved through the good aerodynamic design of the added profile. In his report (Whitcomb 1976), Whitcomb provides general guidelines for designing winglets along with a comprehensive comparison of a planar wing with its vertically and horizontally expanded counterparts. Over years, winglets have evolved and proved more and more effective in increasing flight efficiency. Nonetheless, no unique solution exists and each winglet has to be carefully designed based on the objectives and constraints. Works published by Gerontakos and Lee (2006), Falcão et al. (2011), Gavrilovic et al. (2015), and Oda et al. (2017) present typical parametric studies and optimization procedures related to winglet designs.

The success of winglets in airplanes incites their possible applications in TVC control. It should be, however, noted that reducing induced drag requires the global flow patterns to be modified, and that is why small tip devices, which induce localized changes in the flow, are not so effective for this purpose (Spalart 1998; Kroo 2005; McLean 2005). This is in contrast to TVC where even small changes in the viscous core structure could result in huge advantages by limiting the pressure deficit. Although the concept of vertical tip expansions already exists in hydraulic machines, much less attention has been paid to it. The so-called ‘anti-cavitation lips’ usually have a very simple geometry and are primarily designed to shift TVC away from the blades (Roussopoulos and Monkewitz 2000). We carried out a survey on axial turbines that were tested in EPFL Laboratory for Hydraulic Machines over the last 15 years (more than 40 projects). This survey, which involves major turbine manufacturers, revealed that only 27% of the tested machines were equipped with anti-cavitation lips; all of which were oriented to the suction side of the blades. Regarding this mitigation method, Gim and Lee (2013) study the flow characteristics of a NACA-0018 hydrofoil with an endplate and assert that the plate disturbs the roll-up process but provide no data on the cavitation performance. Ren et al. (2017) evaluate horizontal axis tidal stream turbines and show that a triangle winglet reduces the tip vortex strength and improves the energy conversion, however, no cavitation data are provided. The application of winglets in marine propellers was recently investigated by Ommundsen (2015) and Gao et al. (2019). The former research focuses on the improvement of the propeller performance and not the TVC behavior, while the latter demonstrates that certain winglets can improve the TVC behavior without affecting the propeller performance.

Although the literature highlights promising potentials for winglets, the fact that most of the available cases were performed on complex geometries makes the physical comprehension less intuitive and therefore encourages for more research. In this regard, the current work aims to thoroughly address the effects of non-planar winglets on TVC. For this purpose, an elliptical hydrofoil is selected as the baseline geometry and various winglets are tested. For the sake of simplicity, the winglets are obtained by smooth non-planar extensions of the ending part of the baseline hydrofoil. Therefore, in our study, the design variables consist of the dihedral angle (θ) and the length of the bent section (L_BS). By varying these parameters, a number of new hydrofoils with different winglet configurations are designed and manufactured. Thereafter, TVC inception–desinence tests and flow visualizations are performed on these hydrofoils at various incidence angles and freestream velocities to assess their effectiveness in mitigating cavitation within the tip vortex. Lift (L) and drag (D) forces are also measured to guarantee that the new geometries do not impose a negative effect on the hydrodynamic performances of the hydrofoils. Finally, stereo particle image velocimetry (Stereo-PIV) is employed to measure the non-cavitating velocity field in the wake of the hydrofoils to determine how the vortex parameters are influenced by the winglets.

2 Experimental setup

In the present study, an elliptical hydrofoil with NACA 16-020 cross-section is selected as the baseline geometry. The thickness (y_b) of this symmetrical hydrofoil is given by the following equations, with x_b being the chord-wise position measured from the leading edge. The span (S) and the root-chord (C₀) of the baseline hydrofoil are 90 and 60 mm, respectively:

$$\begin{aligned} & \left\{ {\begin{array}{*{20}l} {\frac{{y_{b} }}{{C_{0} }} = a_{0} \left( {\frac{{x_{b} }}{{C_{0} }}} \right)^{{\frac{1}{2}}} + a_{1} \left( {\frac{{x_{b} }}{{C_{0} }}} \right) + a_{2} \left( {\frac{{x_{b} }}{{C_{0} }}} \right)^{2} + a_{3} \left( {\frac{{x_{b} }}{{C_{0} }}} \right)^{3} } \hfill & {0 \le \frac{{x_{b} }}{{C_{0} }} \le 0.5} \hfill \\ {\frac{{y_{b} }}{{C_{0} }} = b_{0} + b_{1} \left( {1 - \frac{{x_{b} }}{{C_{0} }}} \right) + b_{2} \left( {1 - \frac{{x_{b} }}{{C_{0} }}} \right)^{2} + b_{3} \left( {1 - \frac{{x_{b} }}{{C_{0} }}} \right)^{3} } \hfill & {0.5 \le \frac{{x_{b} }}{{C_{0} }} \le 1} \hfill \\ \end{array} } \right. \\ & a_{0} = + 0.197933\quad a_{1} = - 0.047850\quad a_{2} = - 0.008200\quad a_{3} = - 0.111880 \\ & b_{0} = + 0.002\quad b_{1} = + 0.465\quad b_{2} = - 0.684\quad b_{3} = + 0.292 \\ \end{aligned}$$

The experiments are carried out in the EPFL high-speed cavitation tunnel with a 150 × 150 × 750 mm test section, which is surrounded by Plexiglas windows on three sides providing a good visual access. The maximum velocity at the inlet may reach 50 m/s with turbulence levels as low as 1%, and the static pressure may be varied between vapor pressure and 16 bars. Reynolds number (Re) and cavitation number (σ) are defined by the following relations, in which $U_{\infty }$ and $p_{\infty }$ represent the freestream velocity and static pressure, and $\rho$, $\mu$, and $p_{\text{v}}$ stand for density, dynamic viscosity, and saturation vapor pressure of water, respectively:

$$Re = \frac{{\rho U_{\infty } C_{0} }}{\mu },\quad \sigma = \frac{{p_{\infty } - p_{\text{v}} }}{{0.5\rho U_{\infty }^{2} }}.$$

Various winglet configurations are realized by smooth non-planar extensions of the ending part of the baseline hydrofoil. The length of the bent section is either 5 or 10% of the baseline span and the dihedral angle between the original and the bent planes is either 45° or 90°. In airplane wings, a positive dihedral angle represents a winglet that is tilted up toward the suction side, while a negative dihedral (also called anhedral) angle is attributed to a winglet that is bent down toward the pressure side. As a convention in this paper and in analogy to airplanes where the suction side is always on top, bent-upward and bent-downward terms are designated to positive and negative dihedral angles, respectively. The effect of the dihedral angle is investigated at $L_{\text{BS}} = 0.1S$ with θ = ± 45° and ± 90°. To examine the influence of the bent section length, winglets with L_BS = 0.05S and θ = ± 90° are also tested. We took profit of the symmetry of the baseline profile and produced only one hydrofoil for each θ − L_BS configuration and tested them at positive and negative incidence angles. In our setup, the winglets are bent downward when tested at α > 0, and vice versa.

Two additional hydrofoils are also produced. One hydrofoil truncated at 90% of the span is built and used as the reference case for the lift-and-drag force measurements. In fact, from the aerodynamics point of view, the baseline elliptical hydrofoil is a horizontal extension of this truncated hydrofoil and serves as a remedy for attenuating tip vortex and decreasing lift-induced drag. The second hydrofoil is inspired from the latest trends in civil airplanes and is equipped with a double-sided winglet. The upper and lower fins have similar geometries with $L_{\text{BS}} = 0.1S$ and θ = 45°. The manufactured hydrofoils and their effective spans and projected surface areas are depicted in Fig. 1.

Fig. 1

Top: manufactured hydrofoils from bronze, bottom: top- and side-views of the different hydrofoils with their effective spans and projected lifting surface areas

For TVC inception–desinence tests, we adopt the same procedure as in (Amini et al. 2019). At each flow condition (U_∞ and α fixed), $p_{\infty }$ (and correspondingly σ) is initially set high enough to avoid any cavitation in the test section. The pressure is gradually decreased until a stable cavity attached to the hydrofoil tip forms within the tip vortex, which gives the inception threshold (σ_i). The pressure is further decreased to allow for developed cavitation and thereafter increased to suppress it. The limit where TVC totally disappears is considered as the desinence index (σ_d). The inception–desinence test is repeated twice for each flow condition. The whole process for each point might take up to 10 min, long enough to ensure that flow is provided with sufficient time to react to pressure changes. It should be noted that all the measurements are performed in nearly saturated water (above 90% of O₂ saturation) at 18 °C. The saturation level of water is measured by an oxygen sensor (PreSens O₂ Dipping Probe, DP-PSt3) installed upstream of the test section. The cavitation inception–desinence measurements are conducted at U_∞ = 15 m/s and α = 4°, 6°, 8°, 10°, 12°, 14°, and 16°.

Flow visualization is also used for assessing the winglets’ effectiveness in TVC alleviation. For this purpose, the cavitating flows at α = 12° and U_∞ = 10 and 15 m/s are illuminated by a flash light with a duration of approximately 30 µs and still photographs are taken by a Nikor 60-mm f/2.8D objective.

To measure the hydrodynamic forces, the hydrofoils are mounted on a five-component load cell, which consists of a deformable H-beam and several strain gauges forming five full-bridge circuits. With this balance, the exerted lift and drag forces are measured with precisions of 1.5 N and 0.5 N, respectively (Dupont 1993). In the present research, the force measurements are performed at U_∞ = 10 m/s in non-cavitating flow. For each incidence angle, 2000 samples are acquired over a 2-s period and averaged. To have a fair comparison between the different hydrofoils, lift and drag coefficients are calculated based on the measured forces by the equations below, in which $A_{\text{ref}}$ stands for the projected lifting surface area of each configuration indicated in Fig. 1:

$$C_{\text{L}} = \frac{L}{{0.5\rho U_{\infty }^{2} A_{\text{ref}} }},\quad C_{\text{D}} = \frac{D}{{0.5\rho U_{\infty }^{2} A_{\text{ref}} }}.$$

We employ Stereo-PIV technique for measuring the velocity field in the wake of the hydrofoils in non-cavitating flow. A similar setup to that of Dreyer et al. (2014) is used with slight modifications. The measurement plane for the Stereo-PIV is located at 2.5C₀ downstream of the hydrofoil tip, as shown in Fig. 2. This location is far enough from the hydrofoils, so that the roll-up process is completed and the tip vortex is fully developed. A double-pulsed YAG laser (Litron DualPower, 532-nm wavelength, and 135 mJ/pulse) equipped with a cylindrical lens is used to generate a 1-mm-thick illuminated sheet at the measurement location. A water-box with 30° side walls is attached to the side window of the test section to avoid optical distortions. Two CCD cameras (Dantec FlowSense E0 4 M with resolution of 2048 × 2048 pixels) equipped with Nikkor 60-mm f/2.8D objectives are mounted with a slight tilt to align the focal plane with the laser sheet (the Scheimpflug principle). The seeding particles are made of hollow glass spheres with an average diameter of 10 μm. We acquire 1000 image-pairs for each flow condition, which is adequate for a good estimation of the averaged velocity fields. For calibration purposes, a 100 × 100 mm double-face target with dark markers on a bright background is immersed in the test section. The target is 0.5 mm thick. The diameters of the reference, axis, and main markers are 2, 1, and 1.5 mm, respectively, and the markers are distributed with a uniform spacing of 2.5 mm. The target is pictured from both sides by the PIV cameras at different stream-wise positions about the laser’s mid-plane (0, ± 0.5 mm, ± 1 mm), and thereafter a third-order polynomial is used for fitting the coordinates. It is noted that the final field of view after the stereo mapping is about 60 and 30 mm in x and y directions, respectively.

Fig. 2

Left: top view of the Stereo-PIV setup similar to that of Dreyer et al. (2014), right: the elliptical baseline hydrofoil with the coordinate system

We have implemented a cross-correlation algorithm with 50% overlap to process the PIV images. The interrogation windows were initially set to 128 × 128 pixels and then refined to 32 × 32 pixels, leading to a vector-to-vector resolution of 0.35 mm in x and y directions. Range validation and moving-average validation filters are sequentially applied to the calculated velocity fields. The time interval between the two successive images (Δt) is set in a way that an illuminated particle travels no more than one-fourth of the final interrogation area (which resulted in Δt = 10 μs at U_∞ = 10 m/s for instance). The wandering motion of the tip vortices, i.e., the random fluctuations of the vortex axis, is corrected before averaging the velocity fields. For this purpose, the instantaneous vortex center of each sample is identified by applying the Graftieaux et al.’s (2001) algorithm. The velocity fields are re-aligned accordingly and then averaged. Since the vortex centers do not necessarily lie on the grid nodes, a 2D cubic spline interpolation is performed for the accurate re-alignment of the samples. [For more information, the readers may refer to Dreyer et al. (2014)].

3 Results and discussion

3.1 TVC inception and desinence measurements

Figure 3 shows TVC inception and desinence thresholds of different hydrofoils (with L_BS= 0.1S) obtained at U_∞ = 15 m/s for incidence angles varying between α = 4° and 16°. Apparently, for most of the flow conditions, the winglet-equipped hydrofoils perform better than or equal to the baseline. The differences between the cavitation thresholds of different geometries are more pronounced for incidence angles above α ~ 10°. The 90°-bent-downward (toward the pressure side) winglet is well distinguished from the other geometries by a considerable difference. The green downward-pointing triangles in Fig. 3 stand by far below the other data; for instance at α = 14°, the 90°-bent-downward winglet decreases σ_i from 2.5 for the baseline down to 0.8, equivalent to an enormous reduction of 68%. The 2nd best performance is obtained by the 90°-bent-upward. The other configurations, i.e., 45°-bent-upward and downward and double 45°, lie close to each other and show almost the same or slightly better behaviors than the baseline geometry. On the other hand, it is observed that the truncated hydrofoil is associated with noticeably higher inception/desinence cavitation indices. In fact, the local cavitation that occurs at the sharp edges of this profile at high cavitation numbers provides a sufficient amount of nuclei that advance the TVC inception.

Fig. 3

TVC inception–desinence results of different hydrofoils for various α at U_∞ = 15 m/s

At the desinence limit, the differences between the baseline and the winglet-equipped hydrofoils become even more visible. This is probably due to the fact that the baseline hydrofoil is very prone to delayed desinence leading to large hysteresis, which is due to the formation of a laminar separation bubble that enhances water outgassing (Amini et al. 2019). Comparing the inception and desinence indices in Fig. 3 reveals that hysteresis is significantly reduced by the winglets and even suppressed for θ = ± 90°. This is a clear indication that the added curvature at the tip of the hydrofoil disturbs the laminar separation of the boundary layer, resulting in a reduction of the inception–desinence hysteresis.

TVC inception–desinence results of θ = ± 90° winglets with L_BS = 0.05S are presented in Fig. 4 and compared with those of the 10%-bent 90° winglets as well as the baseline. As is clear in Fig. 4, reducing L_BS to 5% of the baseline span undermines the performance of the 90° winglets, drastically. At the inception limit, the 5%-bent-upward has almost the same behavior as the baseline and no gains are achieved. However, comparing the same pair of hydrofoils at desinence shows that the 5%-bent-upward winglet performs slightly better, especially at α = 12° and 14°. Like the case of L_BS=0.1S, it is observed that the 5%-bent-downward winglet is more effective in mitigating TVC in comparison to the 5%-bent-upward one. However, the TVC desinence thresholds presented in Fig. 4 show that the 5%-bent-downward winglet is at most as good as the 10%-bent-upward configuration.

Fig. 4

Comparison of TVC inception–desinence results of 5%-bent-90° winglets with 10%-bent-90° and the baseline hydrofoils at U_∞ = 15 m/s for various angles of attack

Regarding the inception, a peculiar behavior is observed for the 5%-bent 90°-downward winglet. For all other configurations, cavitation incepts first intermittently within the tip vortex, and a further small pressure reduction results in development of a stable cavity attached to the hydrofoil tip. However, in the case of the 5%-bent 90°-downward hydrofoil, a new flow state is observed: formation of a stable non-attached cavity within the tip vortex at α = 12°, 14°, and 16° for a relatively wide range of cavitation numbers. Two snapshots of the stable non-attached cavity taken at U_∞ = 15 m/s, α = 16° and σ = 1.8 are shown in Fig. 5. This is why we see two distinct inception thresholds denoted for this winglet in Fig. 4. The higher value represents the inception of the stable non-attached cavity and the lower one indicates the stable attachment of the TVC to the hydrofoil tip. With this in mind, one may argue that the 5%-bent-downward winglet performs as well as the 10%-bent-downward in delaying the inception of an attached TVC. However, the formation of a stable non-attached cavity within the tip vortex is accompanied with the same drawbacks, e.g., noise emission and erosion, as an attached TVC is. Therefore, it seems more appropriate to compare this winglet with the other configurations based on its higher inception cavitation numbers indicated in Fig. 4. More details and analyses about this observation are provided in Sect. 3.4.

Fig. 5

Snapshots of the stable non-attached cavity within the tip vortex trailing from the hydrofoil with the L_BS = 0.05 S and θ = − 90° winglet; flow conditions: U_∞ = 15 m/s, α = 16° and σ = 1.8

3.2 Flow visualizations

Top views of cavitating flow over and past the baseline and the winglet-equipped hydrofoils (L_BS= 0.1S) are presented on Fig. 6 for α = 12°, σ = 1.2, and U_∞ = 10 m/s (Re = 600,000). We have deliberately selected a low pressure level to allow for a well-developed TVC and a fair comparison of different hydrofoils. As a result, a leading-edge cavitation, which extends to half chord, is also formed. This result confirms the positive effect of the winglets over the baseline geometry. Moreover, the leading-edge cavitation remains almost unchanged, which is an indication that the flow field outside the tip area is not affected by the winglets.

Fig. 6

TVC visualization for different hydrofoils at α = 12°, σ = 1.2, and Re = 600,000

It is observed in Fig. 6 that with the baseline hydrofoil, a stable cavity with a relatively large diameter forms within the tip vortex. The cavitating vortex trailing from the 45°-bent-upward winglet holds practically the same size and state as the one of the baseline, showing that almost no gain is achieved by this configuration. However, with the 45°-bent-downward winglet, the size of the cavitating tip vortex reduces significantly compared to the baseline. The double-45° winglet splits the principal tip vortex into two weaker co-rotating vortices, with the one trailing from the lower fin slightly stronger and more stable. Although each of the two vortices are weaker than the original one, the TVC inception–desinence results, presented in Fig. 3, did not show any big advantage for this winglet. Figure 6 clearly demonstrates the effectiveness of the 90° winglets over the other configurations. With the 90°-bent-upward winglet, the tip vortex is so attenuated that the cavity within its core does not form a unique body of vapor anymore, but it is more like a queue of nonunified elongated bubbles transported by the flow. Interestingly, with the downward-facing version of the 90° winglet, no cavitation is observed within the tip vortex at all.

Figure 7 exhibits the same hydrofoils at the same angle of attack (α = 12°) and cavitation number (σ = 1.2) as in Fig. 6, but at U_∞ = 15 m/s (Re = 900,000). Comparing Figs. 6 and 7 reveals that increasing U_∞ results in transition from sheet to cloud cavitation for all the tested hydrofoils with a strong interaction with the cavitating tip vortex. Nonetheless, the winglets still prove very effective, and with the severe cavitation transpiring on the blades, the mitigation effects are even more pronounced at the higher velocity of 15 m/s. Indeed, the most effective winglet is again the 90°-bent-downward configuration, which does not allow any cavitation at all to occur within the tip vortex under these flow conditions.

Fig. 7

TVC visualization for different hydrofoils at α = 12°, σ = 1.2, and Re = 900,000

Flow visualizations presented in Figs. 6 and 7 suggest that for both dihedral angles θ = 45° and 90°, the winglets bent toward the pressure side (θ < 0) are more effective in alleviating TVC. This observation could be explained by the interaction of the wake flow and the tip vortex flow (viscous effects). Let us consider the case of θ = 90°: once the winglet is bent toward the suction side, an arbitrary circumference of the tip vortex is (roughly) 75% exposed to the high-energy low-turbulence freestream flow and 25% to the wake flow. Therefore, the tip vortex is expected to be mainly made out of the freestream flow and the roll-up process is facilitated. However, with the winglet bent toward the pressure side, the tip vortex is mainly exposed to the wake flow, which is less energetic (decelerated) and more turbulent (diffusive). Entraining a higher portion of the wake results in a less concentrated tip vortex for the downward winglet. Moreover, it is relatively difficult for the low-pressure wake to follow the surface contour and fill the large area that is created due to the convex curvature of the surface in the case of the downward winglet; which is not a problem for the upward configuration and the freestream flow. Besides the wake flow effect, the pictures presented in Figs. 6 and 7 imply that the entrainment of nuclei from the leading-edge cavity is also facilitated for an upward winglet. Moreover, bending the winglet upward results in the suction side boundary layer being drawn into the vortex core; while for a downward winglet, the boundary layer developed on the pressure side is mainly entrained. On the other hand, it is originally the high-pressure stream that flows toward the low-pressure region. Consequently, a bent-downward winglet obstructs the outboard flow on the pressure side at the early stages, and thereby, disturbs the roll-up process more effectively.

3.3 Lift and drag force measurements

Lift and drag coefficients of the hydrofoils with L_BS = 0.1S measured at U_∞ = 10 m/s in non-cavitating flow are plotted against the incidence angle in Fig. 8. At low incidence angles, no significant differences are observed and the hydrodynamic performances of the hydrofoils are identical. The plateau observed in C_L around 6° ≤ │α│ ≤ 9° is known to originate from the laminar separation of the boundary layer around the mid-chord of the suction side. At higher angles of attack, namely above α = 7°, small discrepancies appear between the various geometries. It is observed that the double-45° winglet generates the highest lift force, whereas the truncated hydrofoil has the lowest lift for most of the incidence angles. Moreover, the lift generated by the baseline is slightly higher than the 90°-bent winglets, which is in agreement with the results obtained for the winglets of a commercial jet aircraft (Takenaka et al. 2008). On the other hand, the lower lift coefficient of the 90°-bent winglets compared to the baseline could be partly attributed to the fact that the effective lifting surface of this hydrofoil is smaller than the reference area indicated in Fig. 1.

Fig. 8

Lift and drag coefficients of different hydrofoils obtained at U_∞ = 10 m/s

As the incidence angle increases further, the flow separates at the leading edge and stall occurs with a drop in the lift and an increase in the drag. Similar behavior is observed for the symmetrical hydrofoils (baseline, truncated and 2 × 45° winglets), for which the stall occurs around α_stall = ± 20° with a hysteresis loop of about 1°–2°. For the non-symmetrical hydrofoils (90° and 45° winglets), the stall is slightly advanced for positive angles of attack (α_stall = 19°) with almost no hysteresis. On the contrary, the stall is delayed for negative angles of attack (α_stall = − 23°) with an enhanced hysteresis loop that extends to almost 4°. Our observations may be explained by the effect of the tip vortex downwash on the suction side, which contributes to maintaining the boundary layer attached. We argue that bending the hydrofoil toward the pressure side (downward winglets) reduces the downwash effect and leads to a decrease in α_stall as well as a reduction in the hysteresis loop. In contrast, the downwash effect is enhanced when the hydrofoil is bent toward the suction side (upward winglets) with an increase in both the stall angle and the hysteresis loop. Besides, the downwash of the upward winglets affects a larger chord of the foil compared to baseline geometry and thereby delays the stall.

It is noted that the force measurements of the shorter winglets (L_BS = 0.05S) did not reveal any differences and they act almost the same as the 10%-bent 90° winglets. Although based on the results presented in Fig. 8, the hydrofoil with the 90°-bent-downward winglet is not the best profile in terms of the hydrodynamic performance, it is still clear that the losses associated with lift generation are relatively small (a few percent) compared to the gains achieved by this winglet in TVC suppression (reflected in Fig. 3). It should be also reminded that in real applications, there is no need to reduce the lifting surface area to create the winglets. Winglets could be designed and manufactured as retrofit modules that are attached to the original geometries. In this fashion, not only would the original hydrodynamic performances of the blades be preserved, but they could also be further improved as in the case of airplane wings.

3.4 Velocity field measurements

The three velocity components of the non-cavitating vortex flow are measured with Stereo-PIV technique at 2.5C₀ downstream of the hydrofoil tip. For each hydrofoil and every flow condition, the wandering motion of the tip vortex is corrected by the detection of the instantaneous vortex centers and their re-alignments. Figure 9 (left) illustrates the instantaneous (blue dots) and the mean vortex center locations for the baseline hydrofoil measured at α = 12° and U_∞ = 10 m/s. The yellow square specifies the mean center location calculated for the original grid (non-interpolated), while the red star shows the mean center location of the interpolated velocity field. With the aid of the tangential velocity vectors plotted in Fig. 9, it is visually clear that the interpolation leads to a more accurate detection of the vortex center.

Fig. 9

Left: instantaneous and mean vortex center locations for the baseline hydrofoil, right: standard deviations of the vortex center locations in x and y directions for different hydrofoils (flow conditions: U_∞ = 10 m/s and α = 12°)

For each condition, Gaussian curves are fitted to the distributions of the vortex centers in x and y directions. The corresponding standard deviations (σ_x and σ_y) are presented in Fig. 9 (right) for the different hydrofoils at α = 12° and U_∞ = 10 m/s. No distinct preference is observed for the wandering direction and the vortex motion is almost identical in x and y directions. Among the hydrofoils, the baseline shows the lowest wandering amplitude while the 90°-downward (10%) winglet owns the highest. It is also noticeable that the downward-facing winglets induce relatively larger wandering motions in comparison to their upward counterparts. Figure 9 (right) also indicates that the standard deviations of the wandering motions are typically smaller than the achieved resolution of the PIV setup (0.35 mm). Under these conditions, the wandering contributes mainly to sub-grid displacements of the vortices and thereby plays the minor role in smoothening the velocity profiles compared to the finite size of the measurement grid. In this regard, Heyes et al. (2004) have also shown that the wandering correction becomes less effective in recovering the original velocity field as the grid size increases. Despite these facts, our analyses have yielded that for these measurements, correcting the wandering is still worthwhile as it may lead to reductions of up to 5% in the viscous core radius of the tip vortices.

The tangential velocity contours of the different hydrofoils obtained at U_∞ = 10 m/s and α = 12° are presented in Fig. 10. These contours clearly put in evidence the positive role of the winglets, especially the downward configurations, in reducing the rotational velocity at the vicinity of the tip vortex axis. The baseline and the truncated hydrofoils along with upward-bent winglet configurations share quite similar contours with larger tangential velocities and more concentrated cores. In contrast, the three downward-facing winglets develop tip vortices which have relatively diffused cores with much smaller rotational velocities. As it is expected from the TVC inception–desinence results and the flow visualizations, the 10%-bent 90°-downward winglet displays the most striking behavior in terms of the tangential velocity alleviation.

Fig. 10

Contours of tangential velocities for the different hydrofoils at U_∞ = 10 m/s and α = 12°

To have a better comparison, the tangential velocities shown in Fig. 10 are averaged azimuthally and thereafter fitted by a Vatistas vortex model (Vatistas et al. 1991) with the least-squares method. The resulting tangential velocity profiles are plotted in Fig. 11 for the baseline hydrofoil and the 90° upward and downward winglets (L_BS = 0.1S). The velocity profiles are normalized against the maximum tangential velocity and the viscous core radius of the baseline hydrofoil ($V_{{\theta_{ \rm {max} } }}$ = 4.028 m/s and $r_{\text{c}}$ = 2.063 mm). As is clear in Fig. 11, the vortex strength remains almost unchanged for the three hydrofoils and the winglets mitigate the tip vortex by diffusing the concentrated vorticity radially, which results in the enlargement of the viscous core and the consequent diminishment of the tangential velocity peak. The 90°-downward winglet presents outstanding suppression effects with increasing the core radius by 70% and decreasing the maximum tangential velocity to almost 50% of the baseline profile.

Fig. 11

Comparison of the azimuthally averaged tangential velocity profiles for the baseline hydrofoil and the 10%-bent 90° winglets at U_∞ = 10 m/s and α = 12°

Figure 12 compares the different hydrofoils in terms of the strength ($\varGamma$) and the viscous core radius ($r_{\text{c}}$) of the tip vortices at α = 12° and U_∞ = 10 m/s. Apparently, the vortex circulations are slightly affected with less than 10% of variations observed between the baseline and the winglet-equipped hydrofoils. Based on the earlier-presented lift and drag force measurements, which did not show any significant discrepancies between the hydrofoils, and the fact that a direct proportionality exists between the generated lift and the tip vortex strength, this becomes an expected observation. Thus, the dominant suppression mechanism has to be the enlargement of the viscous cores, which is well reflected in Fig. 12 (right) where almost all the winglets have larger core radii than the baseline. It is also observed that for the 90° winglets (L_BS = 0.05 S and 0.1 S), the downward configurations lead to remarkably larger core sizes, which is in accord with the TVC inception–desinence test results. Viscous core thickening is an indication of the increased radial diffusion of vorticity due to augmented momentum mixing rates. Similar observations were reported for the case of a squared wingtip (McAlister and Takahashi 1991; Anderson and Lawton 2003; Lee and Pereira 2010) confirming that sharp edges are responsible for local flow separations that lead to messier flows with more diffused vortices with larger viscous cores. Similarly, a tripped boundary layer is also found to increase the viscous core radius (Dreyer et al. 2014). Therefore, it could be concluded that the winglets, especially when facing downward, introduce local disturbances in the flow, which increase the momentum mixing rates, and consequently, lead to smoothened velocity profiles. This observation is also in line with the results presented by Takenaka et al. (2008) who show that for total drag reduction, winglet span and dihedral angle are the key parameters. They report that a longer winglet acts better and that a horizontal extension is superior to a vertical extension, because the junction flow increases the wave drag. Also in this regard, Gerontakos and Lee (2006) affirm that despite reducing the induced drag, winglets resulted in increased total drags in their experiments due to junction flows. Interestingly, they report a much better performance for the downward-facing winglets in terms of induced-drag reduction. In our case, the elliptical hydrofoil results in a clean roll-up process that concentrates the vorticity in a small core; while the sharp geometries associated with the vertical winglets lead to messier flows with diffused vortex cores, which are proved to be effective in TVC suppression.

Fig. 12

Comparison of the different hydrofoils at U_∞ = 10 m/s and α = 12°. Left: tip vortex strengths, right: viscous core radii

The idea of the higher mixing rates is also supported by the axial velocity contours presented in Fig. 13. As one moves from the baseline hydrofoil to the winglets, and in particular the downward configurations, the jet-like area formed around the vortex axis diminishes and even turns into a completely wake-like area in the case of the 10%-bent 90°-downward winglet. It is elaborated in the literature (Lee and Pereira 2010) that a jet-like axial velocity profile is achieved when the shear layers of the freestream flow do not get entrained, and consequently, destroyed by the wake flow. Thus, the wake-like profiles observed in Fig. 13 imply that the downward winglets facilitate the entrainment of the low-energy wake flow, as was suggested by the flow visualizations in Figs. 6 and 7. The increased mixing of the freestream with the turbulent wake is expected to provoke the wake-like axial pattern as well as the observed suppression in the tangential velocity component.

Fig. 13

Contours of axial velocities for the different hydrofoils at U_∞ = 10 m/s and α = 12°

The 4-lobed pattern observed in the axial velocity contours in Fig. 13 is an indication of the second-mode instability of the tip vortex, which is excited most probably due to the merger of a second vortex with the principal one. Flow visualizations (not presented here) suggest that the primary tip vortex merges with either a leading-edge vortex or secondary vortices in the downstream of the tip region similar to those reported in (Arndt et al. 1991). Devenport et al. (1999) show that when two co-rotating vortices merge, the cores are wrapped in a large turbulent region with an intricate double-spiral structure. Arndt et al. (1991) observe some braided structures not only along the cavitating vortex but also in the single phase flow; therefore, suggesting that these structures are not due the instability of the cavity surface but are a result of the interaction of a pair of co-rotating vortices. In this regard, it is shown (Lacaze et al. 2007; Roy et al. 2011; Leweke et al. 2016) that for vortices with axial flow, the second-order azimuthal symmetry wave is the most unstable mode, the excitation of which leads to the formation of a double-helix structure similar to what is seen in Fig. 13 for the axial velocity contours.

Variations of the vortex strength and viscous core radius with the incidence angle at U_∞ = 10 m/s are depicted in Fig. 14 for the baseline hydrofoil and the 90° winglets. Obviously, the vortex strength increases with α; however, as already observed for α = 12°, the values for the different hydrofoils remain close to each other at each incidence angle. In contrast to the vortex strength, the differences in the viscous core radii are more significant: for instance, $r_{\text{c}}$ of the 10%-bent 90°-downward winglet is almost twice the one of the baseline at any angle of attack.

Fig. 14

Comparison of the baseline hydrofoil with the 90° winglet configurations at U_∞ = 10 m/s and various angles of attack. Left: tip vortex strengths, right: viscous core radii

Regarding the viscous core radii, it is observed in Fig. 14 (right) that the trends are conserved for all the hydrofoils except the 90°-downward (5%) winglet; for which $r_{\text{c}}$ begins to shrink considerably as α increases beyond 12°. Interestingly at the same incidence angles, this winglet holds the highest values of $\varGamma$. To better picture the compound effect of $\varGamma$ and $r_{\text{c}}$ on the caused pressure drops, one could use the other non-dimensional form of the vortex strength, i.e., $\varGamma^{*} = \varGamma /r_{\text{c}} U_{\infty }$, for which it could be shown that the minimum pressure coefficient at the vortex axis ($C_{{{\text{p}}_{ \rm {min} } }}$) is proportional to $\varGamma^{{*^{2} }}$ (Dreyer et al. 2014). Accordingly, Fig. 15 exhibits the variations of $\varGamma^{{*^{2} }}$ with the incidence angle.

Fig. 15

Comparison of $\varGamma^{{*^{2} }}$ between the baseline hydrofoil and the 90° winglet configurations for different incidence angles at U_∞ = 10 m/s

As it is clear in Fig. 15, for α ≤ 12°, the 90°-downward (5%) winglet lies between the 10%-bent 90°-downward and the upward winglets; however, at α = 14° and 16°, $\varGamma^{{*^{2} }}$ (and proportionally the magnitude of $C_{{{\text{p}}_{ \rm {min} } }}$) rises drastically for the 90°-downward (5%) hydrofoil and takes values close to those of the baseline. This particular behavior becomes more sensible when it is considered along with the TVC inception–desinence results presented in Fig. 4, which also showed that the cavitation indices of the 90°-downward (5%) winglet started to rise significantly beyond α = 12°. This was accompanied by the observation of a stable suspended cavity that would form within the tip vortex but not attach to the hydrofoil tip for a relatively wide range of pressure variations during the inception phase. This behavior resulted in the introduction of the two-criterion concept for the TVC inception detection of the 90°-downward (5%) winglet. With the velocimetry results in hand (especially those of Fig. 15), one can argue that the tip vortices generated by the 90°-downward (5%) winglet at α = 14° and 16° are strong vortices whose roll-up processes are spatially retarded, and therefore the generated cavity does not initially attach to the hydrofoil tip. This specific winglet might generate local fluctuations in the flow that avoid the formation of a coherent strong tip vortex right after the tip; which are damped out farther downstream. Back to the two-criterion inception, it seems better to rely on the formation of a stable cavity within the tip vortex (and not necessarily attached to the tip) for the 90°-downward (5%) winglet, because this criterion reflects the actual cavitation risk of the fully-developed vortex.

4 Conclusion

The present study aimed to evaluate the effectiveness of non-planar winglets in TVC suppression. For this purpose, an elliptical NACA 16-020 hydrofoil was chosen as the baseline geometry and various winglet configurations were realized by simply bending the terminal part of the span at dihedral angles equal to ± 45° and ± 90°. Cavitation inception–desinence tests and flow visualizations clearly demonstrated that almost for all the flow conditions, the winglet configurations performed better than, or in the worst cases equal to, the baseline hydrofoil in terms of delaying TVC inception. For the same length of the bent section (L_BS = 0.1S), it was observed that θ = 90° yielded much better results in comparison to θ = 45°. Our findings also showed that for the 90°-bent winglets, the negative dihedral angle (i.e., bent down to the pressure side) is notably superior to the positive one (i.e., bent up to the suction side). Moreover, the study on the bending length effect, which was conducted for the 90°-bent winglets, revealed that the longer vertical section (L_BS = 0.1S) outperforms the shorter one (L_BS = 0.05S) in delaying the TVC inception. Furthermore, lift and drag force measurements showed that the hydrodynamic performances of the hydrofoils were not significantly affected by the winglets. In addition, the Stereo-PIV experiments proved that the winglets reduce the risk of TVC by attenuating the pressure deficit, which is principally achieved by thickening the viscous cores. The results indicated that the tip vortex trailing from the 10%-bent 90°-downward winglet has the lowest maximum tangential velocity (almost half of the baseline) and the largest viscous core radius (70% larger than the baseline), leading to a considerable increase in the pressure at the vortex axis. For this winglet, the axial velocity component has a wake-like profile around the vortex core, which is thought to further limit the pressure drop. Our results also imply that the higher effectiveness of downward configurations is due to the fact that a downward-facing winglet facilitates the entrainment of the wake into the vortex flow, which increases the momentum diffusion rates and thereby smooths down the velocity profiles.

The results presented here clearly assert the benefit of non-planar winglets in mitigating TVC. The winglets are shown to remarkably attenuate TVC without imposing any tangible penalties on the hydrodynamic efficiency. This proven potential calls for further research on the implementation of winglets in real applications such as marine propellers and axial hydro turbines. Namely, the results obtained here are in contrast with the common practice in the turbine industry where wingtip devices (anti-cavitation lips) are traditionally directed toward the suction side. This could be explained by the facts that (1) the tested hydrofoils are non-confined and represent marine propellers better than hydro turbine blades that are enclosed in the casing and develop tip leakage vortex instead, and (2) the initial motivation of anti-cavitation lips is to shift the cavitating tip vortex away from the blades rather than suppressing it. In the end, it should be noticed that our promising results stemmed from simple winglets with non-optimized geometries. We expect that more detailed designs should lead to further improvements not only in TVC suppression but also in the hydrodynamic performance of the blades. The design variables may be expanded beyond the winglet length and dihedral angle and include section, planform, chord distribution, toe angle, sweep, twist, etc., the optimal values of which could be obtained through a multi-objective optimization process.