Investigation of cavitation bubble dynamics near an asymmetric hydrofoil Open Access

Cavitation is one of the primary causes of hydromechanical wear, affecting safe operation of hydraulic machinery.^1–3 The jets and shock waves^4–7 generated by cavitation bubbles produce significant erosion of propellers,⁸ turbine blades,⁹ and other components, which can cause noise and vibration.^10–12 Cavitation can occur at various locations on blades in a flow passage, such as the leading edges of turbine blades, the blade roots, the blade tip edges, the gaps between the blade tips and the discharge ring, and the regions near the trailing edges of the blades. It can greatly reduce turbine efficiency, increase turbine vibration, and exacerbate blade wear. The cross-sectional shape of a turbine blade is that of an asymmetric hydrofoil. Therefore, the present paper explores bubble dynamics near an asymmetric hydrofoil.

There have been a number of studies of bubble dynamic behavior occurring in proximity to curved boundaries such as convex walls, particles, and cylinders.¹³ Using the boundary integral method, Malakhov¹⁴ and Aganin et al.¹⁵ numerically investigated bubble dynamics at locations located symmetrically adjacent to a convex wall. Ma et al.¹⁶ and Tomita et al.¹⁷ used high-speed photography to experimentally investigate the relationship between boundary curvature and cavitation bubble dynamics, and quantitatively analyzed the load characteristics during bubble collapse. Shen et al.¹⁸ investigated the bubble collapse dynamics for different hydrofoils using high-speed photography. Bußmann et al.¹⁹ employed a low-dissipation high-order finite element numerical method taking viscosity and surface tension into account to study the dynamic behavior during the collapse of a single cavitation bubble near a wall and revealed the existence of three jetting modes: needle, mixed, and regular jets. Samaniego et al.²⁰ explored the use of deep neural networks as an alternative to the finite element method for the numerical simulation and solution of engineering problems, providing an approach that could be useful for the investigation of cavitation phenomena.

For cavitation bubble behavior in the presence of particles, Wang et al.¹¹ established a Kelvin impulse model based on Weiss’s theorem and investigated the deformation and motion of a bubble close to two particles with different radii. Li et al.²¹ employed high-speed photography to investigate the mechanisms of the deformation and jetting of a bubble near suspended particles. Koch et al.²² employed both numerical and experimental approaches to explore the deformation and jetting behavior of cavitation bubbles above the flat top of a cylinder. The results of the numerical simulations confirmed the experimental observations of fast jets. Zhang et al.²³ experimentally investigated the dynamics of a pulsating bubble near a cylinder. The deformation and jetting of the bubble were influenced significantly by the proximity of the bubble to the cylinder.

Thus, in summary, there have been abundant investigations of the dynamics of cavitation bubbles near curved boundaries. However, research on cavitation phenomena near hydrofoils has focused primarily on cavitation flows, whereas the dynamic behavior of a single cavitation bubble has garnered less attention.

The present paper primarily explores the deformation and interface motion of a cavitation bubble near an asymmetric hydrofoil, and employs the Kelvin impulse theoretical model to investigate spatial distributions near the hydrofoil. Section II introduces the experimental configuration. Section III introduces the Kelvin impulse model. Section IV discusses the effect of bubble position. Section V explores the effect of the hydrofoil’s eccentricity angle. Section VI presents the Kelvin impulse characteristics near the hydrofoil. Section VII summarizes the conclusions.

This section describes the experimental equipment and procedure, together with the definitions of relevant parameters for the experimental system. Figure 1 illustrates the experimental setup, including the high-speed photographic system. The water chamber contains deionized water. The hydrofoil model consists of two parallel glass plates, with a plate spacing of 1.6 mm The camera is an iXcameras i-speed 510, with a frame rate of 100 000 fps selected for the experiment. The laser generator is a Penny-100A-SC, with a wavelength of 532 nm. The delay generator is a DG535, with a delay resolution of 5 ps. Cavitation bubbles are induced by parallel beams emitted from the laser generator. The beams are diverged and converged, ultimately focusing near the hydrofoil located between the two parallel glass plates. The delay generator sends signals to synchronize the computer with the other devices.

Figure 2 shows the definitions of relevant parameters for the experimental model. The asymmetric hydrofoil is located at the origin of coordinates, and the cavitation bubble is in the first quadrant, surrounded by deionized water. The length and thickness of the asymmetric hydrofoil are denoted by l and t, respectively, the upward offset height of the center of the hydrofoil is denoted by h, and the equivalent bubble radius is denoted by R_max. the bubble position angle is denoted by θ. The projections of the hydrofoil and the bubble centroid onto the x axis are denoted by a and a′, respectively. The roundness of the cavitation bubble is denoted by ϕ, which represents the degree to which the cross section of the bubble approximates a circle.

As shown in Fig. 3(b), the relevant terms for the bubble radius are first obtained by solving the radial equation of motion of the cylindrical bubble. Then, the complex potential near the asymmetric hydrofoil is calculated. Finally, the relevant terms are substituted to calculate the Kelvin impulse. The radius of the cavitation bubble is calculated from the measured area of the bubble.

On the one hand, the cavitation bubble maintains a fairly cylindrical shape throughout most of its oscillation process (about 80%), as demonstrated in our previous work²⁴ through side-view images. On the other hand, at the selected moment of cavitation bubble collapse in the theoretical study, the bubble retains a well-defined cylindrical shape. The theoretical results are employed to predict the subsequent deformation trend of the bubble.

Figure 4 presents the variation of the Kelvin impulse angle with dimensionless distance l* near an asymmetric hydrofoil. In this figure, curves of different colors represent the theoretical Kelvin impulse directions corresponding to different values of the hydrofoil’s eccentricity angle, and the corresponding colored dots denote the experimental data on the jet angles of cavitation bubble collapse. As can be seen, the Kelvin impulse angles under different conditions match well with the experimental jet angles, verifying the consistency between the high-speed photography experiments and the Kelvin impulse theoretical model.

The behavior of cavitation bubbles located above a hydrofoil has previously been studied in detail.²² However, the concave section below an asymmetric hydrofoil is also prone to cavitation. Therefore, this paper investigates the dynamic behavior of a cavitation bubble located below a hydrofoil and analyzes the influences of the spatial position of the bubble and the hydrofoil’s eccentricity angle.

This section investigates the influence of the dimensionless distance l* on deformation characteristics and bubble interface motion during the evolution of a bubble for fixed values of δ = 120° and θ = 210°.

Figure 5 displays the phenomenon of cavitation collapse for different values of the dimensionless distance l*. In Fig. 5(a), for l* = 3.57, during the growth of the bubble, it exhibits a clear columnar feature. By contrast, during its collapse, a depression appears on the bubble interface far away from the hydrofoil, eventually evolving into a liquid jet. Under the influence of the jet and the wall, the bubble becomes B-shaped adjacent to the wall. The jet then penetrates the entire cavitation bubble, causing it to divide into two sections that move apart to either side. In Fig. 5(b), for l* = 4.12, the bubble displays distinct cylindrical features during its growth stage. During its collapse, the lower end of the bubble adheres closely to the hydrofoil surface, while a liquid jet is formed on the bubble interface farthest from the hydrofoil. The bubble also forms a B-shape in the later stages of collapse and ultimately splits. In Fig. 5(c), at l* = 4.82, the cavitation bubble also exhibits distinct cylindrical characteristics during its growth stage. However, during the later stages of collapse, a depression emerges on the bubble interface far from the hydrofoil surface and evolves into a liquid jet. The bubble forms a heart shape and then ultimately splits. It can be concluded that as l* increases, the shape of the bubble in the later stages of collapse transitions from a B shape to a heart shape.

Figure 6 illustrates the contours and centroid motion during the collapse of a cavitation bubble near an asymmetric hydrofoil for different values of the dimensionless distance l*. For l* = 3.57 and 4.12 in Figs. 6(a) and 6(b), respectively, the contours of the cavitation bubble closely adhere to the hydrofoil during the collapse stage. Then, a significant depression is observed, transforming the bubble contour into a B shape. The contour deformation in Fig. 6(a) is more pronounced than that in Fig. 6(b). In Fig. 6(c), for l* = 4.82, the bubble contour does not touch the hydrofoil during the collapse stage, and remains largely circular in the pre-collapse stages. However, toward the final stages of the collapse, a depression also emerges, giving the bubble a heart-shape contour. As l* increases, the influence of the hydrofoil diminishes, and the degree of depression decreases. For all the different values of l*, the centroid moves toward the hydrofoil during the collapse.

Figure 7 displays the variations in the motion of characteristic points of a bubble near an asymmetric hydrofoil for different values of the dimensionless distance l*. During the growth and collapse stages, the characteristic points on the bubble interface move in opposite directions. The velocities of the characteristic points moving toward and away from the hydrofoil are denoted by |v_a| and |v_b|, respectively.

In Fig. 7(a), the trends of variation of the velocity of the characteristic points close to the hydrofoil are the same for different values of l*. All of them reach their maximum values quickly after the initial bubble generation, and subsequently decrease steadily. When the bubble attains its peak volume, the velocities of the characteristic points drop to zero, after which they remain basically unchanged. With increasing l*, the velocities of the characteristic points near the hydrofoil during the growth all increase.

In Fig. 7(b), for all three values of l*, the variations in the velocities of the characteristic points far away from the hydrofoil remain basically consistent. They all reach a relatively high value (∼16 m/s) immediately after bubble generation. At T* = 0.48, the velocities of the characteristic points become zero, after which they continue to increase reaching a maximum value (20 m/s).

Figure 8 presents the variations in the cross-sectional roundness of a bubble during its collapse for different values of the dimensionless distance l*. In the early stages of collapse, the roundness decreases slowly, and then begins to decrease sharply at T* = 0.9 until the end of the collapse process. As l* increases, the roundness at a given moments during the collapse also increases.

In this section, the effects of different values of the hydrofoil’s eccentricity angle δ on the deformation properties, bubble interface motion, and cross-section roundness during the evolution of a cavitation bubble near an asymmetric hydrofoil are investigated. The values of l* and θ are fixed at 2.87° and 210°, respectively.

Figure 9 illustrates the phenomena of bubble collapse in the vicinity of an asymmetric hydrofoil for different values of the hydrofoil’s eccentricity angle. As shown in Fig. 9(a), for δ = 120°, the bubble exhibits a distinct cylindrical shape during its growth. During the collapse, the lower bubble interface adheres closely to the hydrofoil surface, while a noticeable depression forms on the bubble interface away from the hydrofoil, which then develops into a liquid jet. The bubble evolves into a B-shaped form in the final stages of collapse and subsequently splits into two parts that move in opposite directions.

As shown in Fig. 9(b), the cavitation bubble also exhibits a distinct columnar feature in the growth stage for δ = 135°. During its collapse, the bubble does not come into contact with the hydrofoil, a depression appears at the bubble interface farthest from the hydrofoil and develops into a weak jet, and the bubble takes a heart shape in the final stage of collapse. As shown in Fig. 9(c), the bubble exhibits a clear columnar feature in its growth stage. In the collapse stage, an arcuate depression forms on the bubble interface farthest from the hydrofoil. With increasing δ, the morphology of the cavitation bubble during its collapse shifts from a B shape to a heart shape.

Figure 10 displays the contours and centroid motion of a bubble collapsing near an asymmetric hydrofoils for three different values of the hydrofoil parameter δ corresponding to those in Fig. 9. As shown in Fig. 10(a), for δ = 120°, the cavitation bubble contour closely follows the hydrofoil surface, with the bubble interface severely deformed by the compression of the hydrofoil. The bubble contour shows a significant indentation under the influence of the jet, taking on a B-shaped form. In Figs. 10(b) and 10(c), for δ = 135° and 150°, respectively, the bubble contour does not touch the hydrofoil surface. During the early collapse stage, the contour remains circular. At the end of the collapse, an indentation appears at the end of the bubble interface far from the hydrofoil surface, and it takes on a heart-shaped form. Figure 10(c) shows less deformation of the bubble contour compared with Fig. 10(b), with a milder indentation of the bubble interface. As δ increases, the effect of the hydrofoil gradually decreases, and the degree of indentation of the bubble interface decreases significantly.

Figure 11 demonstrates the variations in the velocities of characteristic points on a cavitation bubble near an asymmetric hydrofoil for three different values of the hydrofoil parameter δ. In Fig. 11(a), during bubble growth, the velocities of the characteristic points all reach their maximum values immediately after bubble generation, and then decrease continuously as the bubble approaches its peak volume. During bubble collapse, for δ = 120°, the velocities of the characteristic points remain zero. As δ increases, the velocities of the characteristic points also increase. In Fig. 11(b), the velocity trends of the characteristic points far from the hydrofoil wall are similar for all three values of the hydrofoil’s eccentricity angle. They all reach a relatively high value (∼16 m/s) shortly after the initial formation of the bubble, and then steadily decrease as the bubble attains its peak volume. During bubble collapse, the velocities of the characteristic points continue to increase until they reach their maximum values. In particular, for δ = 150°, the velocities of the points far from the hydrofoil are considerably higher than for the other two values of the hydrofoil’s eccentricity angle.

Figure 12 illustrates the changes in the cross-sectional roundness of a cavitation bubble during its collapse near an asymmetric hydrofoil for different values of the hydrofoil’s eccentricity angle δ. For δ = 135° and 150°, the respective trends of variation of the roundness of the bubble during its collapse are largely consistent, with the bubble roundness remaining basically unchanged during the pre-collapse stage until T* = 0.9, at which time the bubble roundness starts to decrease abruptly until the end of collapse. For δ = 120°, the bubble roundness decreases slowly in the initial stages of collapse, until, again at T* = 0.9, the roundness starts to drop sharply until the end of the collapse.

This section investigates the spatial distribution of the Kelvin impulse experienced by a cylindrical bubble near an asymmetric hydrofoil. It explores the effects of the hydrofoil’s eccentricity angle δ on the distribution of the sensitivity index of the Kelvin impulse direction and the variation of the Kelvin impulse angle with dimensionless distance l*.

Figure 13 presents the distribution of H for a cylindrical bubble for different values of the hydrofoil’s eccentricity angle δ = 120°, 135°, and 150°. The white area between the two black lines indicates that when the bubble is in this position, it is affected by the hydrofoil and cannot fully grow. As can be seen, the asymmetric hydrofoil is characterized by a convex wall above and a concave wall below. When positioned above the asymmetric hydrofoil, the bubble experiences the greatest impact in the tail region (0° < θ < 30°), where a high-sensitivity area exists. The head region (140° < θ < 180°) is also significantly affected, featuring another high-sensitivity area, while the middle region (30° < θ < 140°) experiences the least impact. When the bubble is positioned below the asymmetric hydrofoil, it again experiences the greatest impact in the tail region (330° < θ < 360°), accompanied by a high-sensitivity area. The head region (180° < θ < 220°) is also relatively strongly affected, exhibiting a high-sensitivity area. Unlike the position above the asymmetric hydrofoil, when the bubble is in the middle region below (220° < θ < 330°), it experiences the least impact, with the presence of a low-sensitivity area. Furthermore, it can be observed that areas with greater curvature exhibit higher impulse sensitivity.

As δ increases, the curvature of the lower section of the asymmetric hydrofoil decreases, leading to a decrease in the extent of the low-sensitivity area in the middle region below the hydrofoil. However, no significant differences are observed in the high-sensitivity areas in the head and tail regions.

Figure 14 presents the variations of the Kelvin impulse angle θ_k near an asymmetric hydrofoil with respect to l*. As shown in Fig. 14(a), for θ = 90°, the Kelvin impulse angle decreases continuously as l* increases, and its rate of variation also decreases. Furthermore, as δ increases, θ_k increases continuously. As can be seen in Figs. 14(b) and 14(c), for θ = 210° and 345°, the patterns of variation are basically the same. As l* increases, θ_k increases continuously, but its rate of variation again decreases. As δ increases, θ_k also increases continuously.

Investigations of the characteristics of cavitation bubble collapse in the vicinity of an asymmetric hydrofoil have been carried out. The typical collapse morphologies of cavitation bubbles have been explored and three typical collapse scenarios have been identified. The evolution of bubble interface motion has also been analyzed. The distribution of the bubble Kelvin impulse has been analyzed, and the Kelvin impulse has been investigated for different values of the hydrofoil’s eccentricity angle and positional angle.

The following conclusions can be drawn from this study:

1. Cavitation bubbles exhibit three typical morphological evolutions when collapsing in different regions near the hydrofoil, with B-shaped, heart-shaped, and arc-shaped collapses, respectively.

2. As the dimensionless distance l* and the hydrofoil’s eccentricity angle δ increase, the bubble interface velocity adjacent to the hydrofoil gradually increases, while its velocity away from the hydrofoil is less affected.

3. The Kelvin impulse sensitivity is higher in the head and tail regions, and lower in the middle region. Owing to the concave structure below the hydrofoil, a larger region of low sensitivity emerges.

This study has assumed a stationary fluid, but future work could consider cavitation bubble collapse in a flow field. The theoretical approach adopted here can only predict the behavior of cavitation bubbles of cylindrical shape. When a cavitation bubble is in close proximity to a hydrofoil, its deformation becomes significant, and the theory is unable to make predictions in such cases. Therefore, numerical simulation methods will need to be utilized in the future research.

This research was financially supported by the National Natural Science Foundation of China (Project No. 51976056).

The authors have no conflicts to disclose.

Hongbo Wang: Formal analysis (equal); Methodology (equal); Writing – original draft (equal). Cheng Zhang: Conceptualization (equal); Investigation (equal); Methodology (equal). Xinrong Xie: Conceptualization (equal); Software (equal). Junwei Shen: Investigation (equal); Methodology (equal). Yiming Li: Supervision (equal); Visualization (equal). Yuning Zhang: Conceptualization (equal); Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.