Theoretical and experimental research on the impacts of the Joukowsky hydrofoils on the bubble collapse dynamics within a confined space

Abstract

The present paper investigates the bubble collapse dynamics near hydrofoils within a confined space. Experiments involving high-speed photography reveal in detail the typical bubble morphological evolution near different hydrofoils between two glass plates (namely the confined space), and the partitioning of the thickness-related and camber-related parameters is analyzed quantitatively. Based on conformal transformation, the liquid velocity field and Kelvin impulse are used to analyze the bubble collapse characteristics qualitatively and quantitatively, including the bubble interface motion, cross-sectional roundness, and collapse jet. The main conclusions are summarized as follows. (1) The bubble morphological evolution near the hydrofoils can be categorized into five typical collapse shapes, and their partition ranges are significantly affected by the thickness-related and camber-related parameters. (2) The thickness-related hydrofoil parameter positively correlated with the bubble interface motion and cross-sectional roundness, while the camber-related parameter is inversely correlated with them. (3) High-velocity regions between the bubble and the hydrofoil head and tail endpoints explain the bubble interface depressions observed in the experiments.

Nomenclature

Roman letters

$a$

Circle radius (m)

$a^{\prime }$

Radius of the auxiliary circle where the bubble position is located (m)

$c$

Intersection position of the circle and positive x-axis (m)

$d$

Distance between the center of the circle and the origin (m)

$f$

Complex potential of liquid

$h$

Height at which the hydrofoil center deviates upwards (m)

$H$

Cylindrical bubble height (m)

$l$

Length of the hydrofoil (m)

$m$

Point source intensity of cylindrical bubble (${\text{m}}^3\text{/s}$)

$R$

Instantaneous bubble radius (m)

$\dot{R}$

First derivative of $R$ with respect to time ($\text{m/s}$)

$R_{\text{max}}$

Maximum bubble radius (m)

$t$

Time ($\text{s}$)

$u$

Liquid velocity (m/s)

$v$

Bubble interface velocity (m/s)

$w$

Width of the hydrofoil (m)

$T$

Duration of first bubble oscillation period (s)

$z_0$

Bubble coordinate

Greek letters

$\delta$

Angle of the line connecting the center of the circle and the origin ($°$)

$\rho$

Liquid density (${\text{kg/m}}^3$)

${\theta }_b$

Angle on bubble surface ($°$)

${\theta }_k$

Kelvin impulse direction ($°$)

${\theta }_z$

Bubble position angle ($°$)

$\varphi$

Roundness of bubble cross-section

1. Introduction

Cavitation erosion endangers the safety and stability of hydraulic machinery [1], [2], [3], and components such as ship propellers [4], [5] and turbine blades [6], [7] are prone to localized pressure drops during operation, leading to cavitation. At this point, collapse jets and shock waves generated during the cavitation process can impact and erode the hydraulic machinery, resulting in noise, vibration, and severe surface wear [8], [9]. In certain situation such as in vane pumps [10], [11], the flow channels are narrow, and it is difficult to predict the impact of cavitation in such confined spaces. Therefore, the present paper conducts theoretical and experimental research on bubble collapse behavior near hydrofoils (i.e., the cross-sectional shape of the blade) within a confined space.

In hydraulic machinery, boundaries such as flat walls [12], [13], [14], [15], [16], [17], spheres/cylinders [18], [19], [20], [21], [22], [23], [24], [25], [26], and hydrofoils [27], [28], [29], [30], [31], [32] are prevalent and have been investigated widely for their susceptibility to cavitation erosion. Flat walls constitute the most fundamental category of boundary, but complex mechanical internal structures give rise to various boundary types, including single straight walls, parallel walls, and C-shaped walls. For a single straight wall, Lechner et al. [12] investigated the characteristics of the collapse jet under different bubble–wall distances and identified two mechanisms for jet formation: (i) for small distances, the jet is formed by collision of annular liquid flow, resulting in a thin jet with a speed of up to 1000 m/s; (ii) for large distances, the jet is formed by axial focusing, resulting in a wide jet with a speed of ca. 100 m/s. Also, they found that the liquid viscosity is the main reason for the formation of a fast jet near a straight wall. Yin et al. [13] conducted experimental and numerical research on the shock-wave characteristics near a straight wall over multiple periods, and they found that shock waves are generated at the end of each bubble oscillation. Also, via the peak pressure at the wall center, they subjected the destructive effects of the shock waves to quantitative analysis and identified four different stages of how the bubble–wall distance influences the pressure peak. For parallel walls, Rodriguez et al. [14] used numerical simulations to investigate how the wall spacing affects the bubble dynamics. The results indicated that compared to a single wall, the second wall reduces the liquid pressure at the farthest point of the bubble interface, thereby decreasing the collapse intensity. As the spacing between the two walls increases, the bubble volume, the distance of the centroid movement, and the jet velocity gradually approach those for a single wall. For C-shaped walls, Brujan et al. [15] used experiments involving high-speed photography to investigate the bubble dynamics in a C-shaped channel comprising two parallel walls and a vertical wall. The results showed that when the bubble is close to the vertical wall, the collapse jet is oriented toward that wall; however, when the bubble is farther from the vertical wall, a radial jet is generated, leading to bubble splitting. Subsequently, Brujan et al. [16] conducted further numerical simulations and found that the influence of C-shaped walls on bubble dynamics can be considered as the combined influence of several simple walls, including a single wall, parallel walls, and a right-angled wall. The relative influences of these walls ultimately determine the complete dynamic behavior of the bubble within the channel.

Spheres and cylinders are frequently encountered in underwater explosions. Wang et al. [18] investigated bubble deformations and collapse jets near spheres, and they established a theoretical model for analyzing the flow velocity field and the Kelvin impulse; based on the analytical expression for the Kelvin impulse, they also developed a theoretical model for predicting the bubble motion during the first oscillation. Subsequently, Wang et al. [19] conducted further experimental research on bubble dynamic behavior during the second oscillation, and they analyzed the variations in the bubble equivalent radius and morphological evolution. Borkent et al. [20] used experimental and numerical methods to investigate the acceleration effect of a bubble on a sphere, and the results indicated that the differential pressure resulting from the bubble growth is the primary factor for the sphere motion. In addition to the acceleration effect, Gonzalez-Avila et al. [21] found that during the bubble oscillation process, spheres farther away from the bubble are subjected to an attractive force, and the small shear force caused by the bubble growth leads to rotation of the sphere. For cylinders, Brett and Yiannakopolous [22] used high-speed imaging and pressure detection technology to investigate the multiple loads on a cylinder caused by bubble oscillation; they found that bubble collapse causes the strongest structural load on the cylinder, followed by shock waves and then bubble pulsation. Pan et al. [23] investigated the bubble dynamics near cylindrical steel wires experimentally and numerically, and the results showed that at different bubble–wall distances, the bubble exhibits three typical collapse cases: (i) flat 2D necking collapse, (ii) teardrop-shaped collapse, and (iii) spherical collapse. Also, the collapse jets weaken with increasing bubble–wall distance until it disappears.

Investigations of bubble dynamics near hydrofoils have focused mainly on cavitating flow and gap cavitation. Huang et al. [27] reviewed the flow structures and mechanisms of unsteady cavitating flow, and taking the Clark-Y hydrofoil as an example, they introduced four types of attached cavitating flow patterns: (i) supercavitation, (ii) cloud cavitation, (iii) sheet cavitation, and (iv) inception cavitation. They also introduced different types of vortical cavitating flow patterns with the Tulin hydrofoil, i.e., developed supercavitation, mixture supercavitation, cloud cavitation, vortex cavitation, and incipient cavitation. Mousavi and Roohi [28] investigated the effects of surface wettability on the supercavitating flow structure near the Clark-Y hydrofoil; they found that cavitation can be delayed by superhydrophobic surfaces at the trailing edge and pressure side or superhydrophilic surfaces at the leading edge. For gap cavitation, Sentyabov et al. [29] used experimental and numerical methods to investigate cavitation in the gap between a hydrofoil end face and a channel; they found that cavitation manifests as gas films within the gap, and the angle of attack of the hydrofoil significantly affects the gas-film formation position. Nichik et al. [30] conducted experimental research on gap cavitation at the tip of a hydrofoil, and they found that it is more severe with a smaller incident angle and a wider gap.

Scholars have also made progress in the research on the bubble dynamics within a confined space. Zeng et al. [33] investigated the bubble jet dynamics within a confined space formed by two rigid walls. They observed three typical jet behaviors under different nucleation positions of the bubble. Furthermore, they found that the gap spacing and the bubble position significantly affect the jet velocity. Azam et al. [34] further conducted high-speed photography experiments from dual perspectives on bubbles within a confined space, investigating the influence of the gap spacing on the bubble dynamics. They found that as the gap spacing increases, the direction of bubble collapse shifted from towards the center of the gap to towards both sides of the gap. Zwaan et al. [35] conducted experimental research on the dynamic behavior of the bubbles near a flat wall within a confined space, and observed typical cylindrical bubble shapes, namely pancake-like structures. They also experimentally observed the bubble collapse jets in rectangular, triangular, and quadrilateral channels. Shen et al. [36] developed a theoretical model for the Kelvin impulse of bubbles near a cylinder within a confined space. They observed various cases of the collapse jets with different intensities and collapse deformations.

In summary, research to date on cavitation near hydrofoils has focused mainly on cavity characteristics, and investigating the dynamics of a single bubble near hydrofoils remains in its preliminary stage. In addition, the effects of the hydrofoil parameters on the bubble behavior have yet to be elucidated. Therefore, the present paper explores the mechanisms by which hydrofoil parameters affect bubble dynamics within a confined space. Based on experiments involving high-speed photography, the typical bubble morphological evolution near a hydrofoil is described in detail, and the theoretical liquid velocity field and Kelvin impulse are used to analyze the characteristics of the bubble collapse both qualitatively and quantitatively. Sec. II introduces the theoretical method and the experimental setup, Secs. III–V analyze the bubble collapse behavior in the tail, middle, and head regions near the hydrofoil, and finally Sec. VI summarizes the conclusions.

2. Methodology

A theoretical model for predicting the velocity distribution of the liquid flow field and the Kelvin impulse of the bubble is introduced. And, the experimental platform and process are demonstrated in detail.

2.1. Theoretical method

In previous work [37], we derived in detail the Kelvin-impulse theoretical model for a cylindrical bubble near a symmetric or asymmetric hydrofoil. Therefore, in this subsection, we present the physical hydrofoil model and the core theoretical formulas of Kelvin impulse theory. Because the present paper investigates two-dimensional flow in a confined space, a complex-plane coordinate system is constructed, and the liquid flow is assumed to be inviscid, incompressible, and potential flow. Fig. 1, Fig. 2 shows a schematic of the model parameters and the conformal transformation, where $z_0$ is the bubble coordinate, $R_{\max }$ is the maximum bubble radius, and ${\theta }_z$ is the bubble position angle. The center of the hydrofoil is at the origin, and its geometry in the coordinate system is expressed as [38]

$$y=\sqrt{\frac{l^2}{4}\left(1+\frac{l^2}{16h^2}\right)-x^2}\pm \frac{2\sqrt{3}}{9}w\left(1-\frac{x}{2c}\right)\sqrt{1-{\left(\frac{x}{2c}\right)}^2}-\frac{l^2+8h^2}{8h}$$ (1)

where

$$h=2d\sin \delta$$ (2)

$$l=4c$$ (3)

$$w=-3\sqrt{3}d\cos \delta$$ (4)

$$a=d\left(4-\cos \delta \right)$$ (5)

Here, h is the height at which the hydrofoil center deviates upward, d is the distance between the center of the circle and the origin in plane 2, $\delta$ is the angle of the line connecting the center of the circle and the origin in plane 2, l and w are the length and width of the hydrofoil, respectively, c is the intersection position of the circle and positive x-axis in plane 2, a is the circle radius in plane 2, and $a^{\prime }$ is the radius of the auxiliary circle passing through the bubble location.

Fig. 1.

Schematic of conformal transformation. Green region: hydrofoil and its transformed circles. Blue region: bubble.

Fig. 2.

Schematic of model parameters.

As shown in Fig. 2, three symmetric and three asymmetric hydrofoils are selected. For the three symmetric hydrofoils (with $\delta$ = 180°), we have $\epsilon$ = 0.13, 0.27, and 0.45, where $\epsilon$ is defined as

$$\epsilon =\frac{d}{c}$$ (6)

and for the three asymmetric hydrofoils (with$\epsilon$ = 0.27), we have $\delta$ = 120°, 135°, and 150°.

To obtain the liquid velocity potential around the hydrofoil and the bubble, the hydrofoil must be transformed into a circular boundary via conformal transformation. Specifically, transformation step 1 is expressed as [38]

$$z^{\prime }=\frac{z+h}{2}+\sqrt{{\left(\frac{z+h}{2}\right)}^2-c^2}$$ (7)

where z and $z^{\prime }$ are the coordinates of any point in planes 1 and 2, respectively. Then, transformation step 2 is expressed as

$$z^{″}=z^{\prime }-d\left(\cos \delta +\text{i}\sin \delta \right)$$ (8)

where $z^{″}$ is the coordinate of any point in plane 3. Also, given the confined space, the bubble grows as a cylindrical bubble, so we consider only its radial motion and neglect heat and mass transfer. At this point, the bubble can be considered as a 2D point source, and the liquid complex potential in plane 3 is obtained from the circle theorem as [38]

$$f\left(z^{″}\right)=\frac{m}{2\pi h}\ln \left(z^{″}-z_0^{″}\right)-\frac{m}{2\pi h}\ln \left(z^{″}-\frac{a^2}{\overline{z_0^{″}}}\right)+\frac{m}{2\pi h}\ln z^{″}$$ (9)

where

$$m=2\pi R\dot{R}H$$ (10)

Here, $z_0^{″}$ is the bubble coordinate in plane 3, $\overline{z_0^{″}}$ is the conjugate of $z_0^{″}$, m is the intensity of the point source and point sink, R is the cylindrical bubble radius, $\dot{R}$ is the first derivative of R, and H is the height of the cylindrical bubble. In Eq. (9), the first, second, and third terms on the right-hand side correspond to the effects of the bubble itself, the point sink, and the point source, respectively. Thus, the additional complex potential dominated by the hydrofoil is expressed as

$$f_{\mathit{add}}\left(z^{″}\right)=-\frac{m}{2\pi h}\ln \left(z^{″}-\frac{a^2}{\overline{z_0^{″}}}\right)+\frac{m}{2\pi h}\ln z^{″}$$ (11)

By substituting Eqs. (7) and (8) into Eqs. (9) and (11), the complex potential in plane 1 is solved for, and the liquid velocity u is obtained as

$$u=\text{Re}\left[\frac{\text{d}f\left(z\right)}{\text{d}z}\right]$$ (12)

Furthermore, the cylindrical bubble Kelvin impulse is expressed as [37], [39]

$$I=-4\pi H\rho {\int }_0^T{\left\{\mathrm{\nabla }\text{Re}\left[F_{\mathit{add}}\left(z\right)\right]\right\}}_{z_0}R\dot{R}dt$$ (13)

where $\rho$ is the liquid density, t is time, and T is the period of the first bubble oscillation. The terms related to the bubble radius can be obtained from the cylindrical bubble wall motion equation [40].

Based on the above theoretical model, the subsequent theoretical results and figures are obtained using a MATLAB program. Fig. 3 shows the flowchart of the MATLAB program.

Fig. 3.

Flowchart of the MATLAB program.

2.2. Experimental setup

The model researched in the present paper is placed between two parallel glass plates that are large enough to allow consideration of an infinite 2D flow field. The spacing between the glass plates is 1.5 mm, much smaller than the bubble diameter (ca. 4 mm), so the bubble is restricted in its axial motion and mainly exhibits radial motion. Fig. 4 shows the detailed layout of the experimental platform together with the models and parameters of the devices. Specifically, the optical axis of the camera, with a speed of 100 000 fps, is normal to the glass plates, and its background illumination is provided by the infrared light source. The laser, with a wavelength of 532 nm and a pulse width of 10 ns, is incident horizontally and focused at the middle position between the two glass plates. The sideview of the model is also exhibited in the figure. Fig. 5 shows a flowchart of the experimental process. In addition, the dimensionless parameters are defined as follows:

$$X^{\ast }=\frac{x}{R_{\text{max}}}$$ (14)

$$Y^{\ast }=\frac{y}{R_{\max }}$$ (15)

$$T^{\ast }=\frac{t}{T}$$ (16)

$$l^{\ast }=\frac{a^{\prime }-a}{R_{\max }}$$ (17)

Here, $R_{\max }$ is the maximum bubble radius, t is time, T is the period of the first bubble oscillation, a is the circle radius in plane 2, and $a^{\prime }$ is the radius of the auxiliary circle passing through the bubble location.

Fig. 4.

Experimental platform configuration and device parameters.

Fig. 5.

Flowchart of experimental process.

Then, the errors in the main parameters involved in the experimental analysis are presented. The errors in x and y directions ($x_{\mathit{err}}/y_{\mathit{err}}$) are equivalent to the width of one pixel, which is $\pm 0.052\text{mm}$. Thus, the error in distance can be calculated as follows:

$$d_{\mathit{err}}=\pm \sqrt{x_{\mathit{err}}^2+y_{\mathit{err}}^2}=\pm 0.074\text{mm}$$ (18)

And, the error in bubble radius ($R_{\mathit{err}}$) is equivalent to $d_{\mathit{err}}$. The error in bubble interface velocity is given as follows:

$$\mathrm{\Delta }{v}_{\mathit{err}}=\pm \frac{d_{\mathit{err}}}{\mathrm{\Delta }t}=\pm 1.8\text{m/s}$$ (19)

Here, $\mathrm{\Delta }t$ is the time interval for calculating the velocity, which is 40 μs. The collapse jet angle is obtained by selecting a bisector of an angle with a length of $2R_{\max }$ in the direction of the jet and measuring its angle. Thus, the error in collapse jet angle is given as follows:

$${\theta }_{\mathit{err}}=\arctan \frac{x_{\mathit{err}}}{2R_{\max }}=\pm 0.74°$$ (20)

3. Bubble collapse behavior in tail region

A hydrofoil can be considered as a complex curved boundary, with its surface being the combination of an angular region and other curved regions with varying curvatures. Thus, when bubbles collapse near different regions of a hydrofoil, various morphological evolution characteristics are observed. Herein, to show the bubble collapse behavior, we divide a hydrofoil into three typical regions: (i) the tail region (with bubble position angle $0°<{\theta }_z<20°$), (ii) the middle region ($20°<{\theta }_z<135°$), and (iii) the head region ($135°<{\theta }_z<180°$). In this section, we investigate the bubble collapse behavior in the tail region in detail, showing typical cases of the bubble morphological evolution and analyzing quantitatively their parameter partitioning near symmetric and asymmetric hydrofoils.

Fig. 6 shows the typical experimental phenomena of the bubble collapse in the tail region with ${\theta }_z=15°$ (Multimedia available online). In each case, the red triangle marks the initial bubble centroid position, and $l^{\ast }$ is the dimensionless bubble–hydrofoil distance. Three typical cases of the bubble collapse are shown. In each image, the thicker black contour is the bubble, and the angular contour in the bottom left is the hydrofoil. The color in the middle of the bubble is consistent with the ambient color, which means that the bubble only exhibits radial motion during most of the oscillation process. Also shown are the time, serial number, scale, and values of the dimensionless parameter. For each case, the morphological evolution process of the bubble during the first oscillation is shown in detail, including the moment of bubble inception, the growth process, the moment of maximum volume, and the collapse process. During the growth process, the bubble basically grows in a circular shape. During the collapse process, when $l^{\ast }$ is relatively small, significant bubble deformations are observed. Specifically, the bubble interface closest to the tip of the hydrofoil first produces a depression, then the depression expands rapidly, causing the deformation and centroid movement of the bubble. As $l^{\ast }$ increases, the bubble deformation and centroid movement both weaken.

Fig. 6.

Typical experimental phenomena of bubble collapse in tail region with ${\theta }_z=15°$ (Multimedia available online): (a) arcuate collapse; (b) elliptical collapse; (c) circular collapse.

Based on the bubble morphological characteristics, three typical cases are distinguished: (i) arcuate collapse, (ii) elliptical collapse, and (iii) circular collapse. The characteristics of these cases are summarized below.

In Fig. 6(a), the bubble exhibits a typical arcuate collapse with a small dimensionless bubble–hydrofoil distance ($l^{\ast }=2.31$). During the collapse process, the bubble interface closest to the tip of the hydrofoil first shrinks and then drives the entire right side of the bubble interface to shrink to the left. At this point, the left bubble interface still maintains an arc shape, while the right interface shrinks and becomes relatively straight. Thus, a typical arcuate collapse is observed in Fig. 6(a-8). In addition, the bubble deformation process and trajectory of the centroid motion are generally counterclockwise. At the end of the collapse, the bubble collapses into a tadpole shape [i.e., Fig. 6(a-10)].

In Fig. 6(b), the bubble exhibits a typical elliptical collapse with a medium dimensionless bubble–hydrofoil distance ($l^{\ast }=3.10$). During the collapse process, the right side of the bubble interface also shrinks significantly but in an arc shape. Thus, a typical elliptical collapse is observed in Fig. 6(b-8). Meanwhile, the bubble centroid movement weakens. At the end of the collapse, the bubble also collapses into a tadpole shape.

In Fig. 6(c), the bubble exhibits a typical circular collapse with a large dimensionless bubble–hydrofoil distance ($l^{\ast }=3.89$). During the collapse process, the bubble maintains a circular shape. From observing the relative position between the red triangle and the bubble, the bubble deformation and centroid movement are the weakest in this case.

The anisotropic characteristics are demonstrated further by recording the motion of the bubble contour. Fig. 7 shows the evolution of the bubble contours over time. In the figure, five moments are selected to show the evolution of the bubble contour; the time intervals between them are roughly the same, so the inter-contour distance indicates the contraction speed. In Fig. 7(a), at the beginning of the collapse, the bubble interface is moving the fastest at the lower right. As the collapse progresses, where the bubble interface is moving the fastest shifts to the upper right; this causes the bubble centroid to move counterclockwise and by the greatest distance in this case. As $l^{\ast }$ increases [i.e., from Fig. 7(a) to (c)], the bubble deformation weakens and the distance moved by the bubble centroid decreases.

Fig. 7.

Evolution of the bubble contours over time. (a) arcuate collapse; (b) elliptical collapse; (c) circular collapse.

Next, we analyze the velocity distribution of the liquid flow field theoretically. Fig. 8 shows the distribution of the liquid velocity in the tail region at a dimensionless time ($T^{\ast }$) of 0.75. At that moment in the experiment, the bubble had not deformed significantly, which satisfies the theoretical assumptions. Thus, the theoretical results for the velocity field are meaningful and can be used to predict the bubble interface motion in the subsequent process.

Fig. 8.

Distribution of liquid velocity in tail region: (a) l* = 1.61; (b) l* = 2.32; (c) l* = 3.03.

In Fig. 8(a), the bubble is extremely close to the hydrofoil, and so a region of high-velocity liquid is observed between the bubble and the hydrofoil tip; the high velocities elsewhere around the bubble are caused by its own oscillatory motion. Around the high-velocity regions, the liquid velocity decreases in the outward direction. When the bubble is at a certain distance from the hydrofoil [e.g., Fig. 8(b)], a low-velocity region is observed between them. As can be predicted, the distribution of the high-velocity region on the bubble surface will significantly affect the direction and degree of the bubble interface depression. Thus, in Fig. 8(b), during the subsequent collapse process at this moment, the lower right side of the bubble will first produce a depression, and according to the distribution of the liquid velocity around the bubble, it will contract generally counterclockwise, which corresponds to the bubble morphological evolution in Fig. 6(a). In addition, as $l^{\ast }$ increases, the low-velocity region expands and the high-velocity region shrinks, which also corresponds to the weakening of the bubble deformation in Fig. 6.

Many experiments were conducted to investigate the bubble collapse near different symmetric and asymmetric hydrofoils. The parameter partitioning of the three aforementioned typical collapse cases is affected significantly by the hydrofoil parameters, and we show quantitatively how the thickness-related parameter ($\epsilon$) for symmetric hydrofoils and the camber-related parameter ($\delta$) for asymmetric hydrofoils affect the parameter partitioning. The shapes of the hydrofoils under different parameter values are shown in Fig. 2.

Fig. 9, Fig. 10 show the parameter partitioning for the three cases near symmetric and asymmetric hydrofoils, and the photographs in the red dashed boxes show the typical bubble collapse morphology associated with l*. As $\epsilon$ increases (i.e., the thickness of the symmetric hydrofoil increases), so does the partition parameter $l^{\ast }$ for the three typical cases. Also, as $\delta$ increases (i.e., the camber of the asymmetric hydrofoil increases), the partition parameter $l^{\ast }$ for cases 1 and 2 decreases, while that for cases 2 and 3 increases.

Fig. 9.

Parameter partitioning for three cases near symmetric hydrofoils: (a)$\epsilon$ = 0.13; (b)$\epsilon$ = 0.27; (c)$\epsilon$ = 0.45. The photographs in the red dashed boxes show the typical bubble collapse morphology associated with l*.

Fig. 10.

Parameter partitioning for three cases near asymmetric hydrofoils: (a) $\delta$ = 120°; (b) $\delta$ = 135°; (c) $\delta$ = 150°. The photographs in the red dashed boxes show the typical bubble collapse morphology associated with l*.

Next, we analyze the liquid velocity around the bubble interface quantitatively. Fig. 11 shows the theoretical results, where ${\theta }_b$ is the angle on the bubble surface as defined in Fig. 11(a). The maximum liquid velocity around the bubble interface occurs at $300°<{\theta }_b<330°$, which is the area on the bubble interface close to the hydrofoil tip. The minimum liquid velocity around the bubble interface occurs around ${\theta }_b=240°$, which is the area on the bubble interface closest to the hydrofoil surface. As $\epsilon$ increases, the maximum liquid velocity increases and the minimum one decreases, while the effect of $\delta$ is the opposite. Thus, the difference in the velocity distribution of the liquid on the bubble interface is the main reason for different bubble collapse morphologies. As $l^{\ast }$ increases, the difference in liquid velocity decreases, leading to a reduction in the deformation degree of the bubble, which ultimately results in a circular collapse.

Fig. 11.

Theoretical results for liquid velocity around bubble interface: (a) symmetric hydrofoils with$\epsilon$ = 0.13, 0.19, and 0.27; (b) asymmetric hydrofoils with $\delta$ = 135°, 150°, and 165°.

Because of the significant bubble deformation in the tail region, we analyze the roundness of the bubble cross section here. Fig. 12 shows the roundness ($\varphi$) of the bubble cross section during the collapse process with $l^{\ast }=3.15$, where $T^{\ast }$ is the dimensionless period of the first bubble oscillation, and $\varphi$ is the degree to which the cross section approaches a circle ($\varphi =1.0$) and is defined as

$$\varphi =\frac{4\pi S}{C^2}$$ (21)

where S is the area and C is the perimeter. As shown in Fig. 12, during most of the bubble collapse process ($T^{\ast }<0.9$), $\varphi$ basically remains relatively large ($\varphi >0.85$), indicating that the bubble deformation is not yet significant. Then, as the collapse proceeds, $\varphi$ decreases sharply. At the end of the collapse, $\varphi$ increases with increasing $\epsilon$ and decreasing $\delta$. Fig. 13 shows the roundness ($\varphi$) of the bubble cross section during the collapse process with $l^{\ast }=2.8$. In the figure, at a smaller $l^{\ast }$, the trend of roundness varying with hydrofoil parameters is consistent with Fig. 12. However, for symmetric hydrofoils, the roundness at the end of the collapse is greater at smaller $l^{\ast }$, whereas for asymmetric hydrofoils, the opposite is true.

Fig. 12.

Roundness ($\varphi$) of bubble cross section during collapse process with $l^{\ast }=3.15$: (a) symmetric hydrofoils with$\epsilon$ = 0.13, 0.27, and 0.45; (b) asymmetric hydrofoils with $\delta$ = 120°, 135°, and 150°.

Fig. 13.

Roundness ($\varphi$) of bubble cross section during collapse process with $l^{\ast }=2.8$: (a) symmetric hydrofoils with$\epsilon$ = 0.13, 0.27, and 0.45; (b) asymmetric hydrofoils with $\delta$ = 120°, 135°, and 150°.

4. Bubble collapse behavior in middle region

In the middle region, the bubble collapse is reflected mainly in the collapse jet and the corresponding bubble deformations caused by the jet.

Fig. 14 shows the typical experimental phenomena of the bubble collapse in the middle region with ${\theta }_z=90°$ (Multimedia available online); again, the red triangle marks the initial bubble centroid position. Two typical cases of the bubble collapse are shown. During the collapse process in the middle region, there is always a depression in the bubble interface far from the hydrofoil, and this depression develops into a jet toward the hydrofoil surface. As $l^{\ast }$ increases, the depression weakens. At the end of the collapse, the bubble is split into two parts by the collapse jet. In addition, the bubble centroid movement is also significant. Based on the bubble-deformation and collapse-jet characteristics, two typical cases are distinguished: (i) “B”-shaped collapse and (ii) heart-shaped collapse. The primary characteristics of these two cases are summarized below.

Fig. 14.

Typical experimental phenomena of bubble collapse in middle region with ${\theta }_z=90°$ (Multimedia available online): (a) “B”-shaped collapse; (b) heart-shaped collapse.

In Fig. 14(a), the bubble exhibits a typical “B”-shaped collapse with a small dimensionless bubble–hydrofoil distance ($l^{\ast }=1.87$). Because of the closeness of the bubble and hydrofoil, the lower side of the bubble interface comes into contact with the hydrofoil during the collapse process, and the collapse jet develops from the upper side, penetrates the entire bubble, and contacts the hydrofoil. At this point, a typical “B”-shaped collapse is observed in Fig. 14(a-8). At the end of the collapse, the bubble is split into two parts by the collapse jet. In addition, the bubble centroid moves toward the surface of the hydrofoil.

In Fig. 14(b), the bubble exhibits a typical heart-shaped collapse with a large dimensionless bubble–hydrofoil distance ($l^{\ast }=3.09$). The bubble does not come into contact with the hydrofoil. Thus, as the collapse jet develops, the lower side of the bubble is still in an arc shape, and a typical heart-shaped collapse is observed in Fig. 14(b-8). In addition, the bubble is also penetrated by the jet at the end of the collapse, and the centroid also moves significantly toward the hydrofoil.

Next, we investigate the anisotropic characteristics via the motion of the bubble contour. Fig. 15 shows the evolution of the bubble contours over time. Five moments are selected to show the evolution of the bubble contour, and the time intervals between them are roughly the same. The most significant movement of the bubble interface occurs in the part far from the hydrofoil, and as the collapse proceeds, the movement distance increases and develops into a jet. For the bubble interface close to the hydrofoil, its movement is minimal. In addition, the direction of the collapse jet is basically consistent with the movement direction of the bubble centroid. As $l^{\ast }$ increases [i.e., from Fig. 15(a) to (b)], the difference in the evolution of bubble morphology is relatively small, but there is a slight deviation in the jet direction, which we analyze quantitatively below.

Fig. 15.

Evolution of the bubble contours over time. (a) “B”-shaped collapse; (b) heart-shaped collapse.

We also analyze the theoretical results for the liquid velocity field. Fig. 16 shows the distribution of the liquid velocity in the middle region at a dimensionless time ($T^{\ast }$) of 0.75. There are low-velocity regions between the bubble and the hydrofoil, and the other parts of the bubble surface are surrounded by a high-velocity region caused by its own oscillatory motion. The highest velocity is observed at the bubble interface away from the hydrofoil, which explains the bubble interface depression in this region in the experiments. As $l^{\ast }$ increases, the low-velocity region expands and the high-velocity region shrinks, which also corresponds to the weakening of the bubble deformation and collapse jet in Fig. 14.

Fig. 16.

Distribution of liquid velocity in middle region: (a) l* = 1.61; (b) l* = 2.32; (c) l* = 3.03.

We also analyze how the thickness-related parameter ($\epsilon$) for symmetric hydrofoils and the camber-related parameter ($\delta$) for asymmetric hydrofoils affect the parameter partitioning in the middle region. Fig. 17, Fig. 18 show the parameter partitioning for the two cases near symmetric and asymmetric hydrofoils. The photographs in the red dashed boxes show the typical bubble collapse morphology associated with l*. As $\epsilon$ increases, so does the partition parameter $l^{\ast }$, and as $\delta$ increases, $l^{\ast }$ decreases. From the evolution of bubble morphology shown in the photographs, it can be observed that the direction of the collapse jet deviates significantly with increasing l*.

Fig. 17.

Parameter partitioning for two cases near symmetric hydrofoils: (a)$\epsilon$ = 0.13; (b)$\epsilon$ = 0.27; (c)$\epsilon$ = 0.45. The photographs in the red dashed boxes show the typical bubble collapse morphology associated with l*.

Fig. 18.

Parameter partitioning for two cases near asymmetric hydrofoils: (a) $\delta$ = 120°; (b) $\delta$ = 135°; (c) $\delta$ = 150°. The photographs in the red dashed boxes show the typical bubble collapse morphology associated with l*.

The Kelvin impulse is effective for predicting the characteristics of bubble collapse jets [37], [39], so here we analyze and compare the variation trends of the collapse jet angle via the Kelvin impulse. Fig. 19 compares the experimental collapse-jet angle and the theoretical Kelvin-impulse angle (${\theta }_k$). The solid lines are the theoretical results for the Kelvin-impulse direction, the dotted lines are the experimental results for the collapse-jet angle, and the different colors represent hydrofoils with different parameters. As shown in Fig. 19, the theoretical Kelvin-impulse direction agrees well with the experimental jet angle. Specifically, ${\theta }_k$ decreases with increasing $\epsilon$ and $\delta$ and increases with increasing $l^{\ast }$. Therefore, the Kelvin impulse acting on the bubble is the main reason for its heart-shaped collapse. When the bubble is close to the hydrofoil, it is restricted by the wall surface, resulting in a “B”-shaped collapse.

Fig. 19.

Comparison between experimental collapse-jet angle and theoretical Kelvin-impulse angle (${\theta }_k$): (a) symmetric hydrofoils with$\epsilon$ = 0.13, 0.27, and 0.45; (b) asymmetric hydrofoils with $\delta$ = 120°, 135°, and 150°.

5. Bubble collapse behavior in head region

In this section, we explore the bubble collapse behavior in the head region and distinguished it into three typical cases of the bubble morphological evolution. We also analyze the characteristics of the bubble interface velocity quantitatively.

Fig. 20 shows the typical experimental phenomena of the bubble collapse in the head region with ${\theta }_z=165°$ (Multimedia available online). Again, the red triangle marks the initial bubble centroid position. Three typical cases of the bubble collapse are shown. In this region, significant collapse jets and bubble interface depressions occur only when the dimensionless bubble–hydrofoil distance ($l^{\ast }$) is small. At the end of the collapse, the phenomenon of the bubble being split by the jet still exists. Meanwhile, the bubble centroid movement is also the most significant in this case. As $l^{\ast }$ increases, the collapse jet and bubble interface depression become weaker and eventually vanish, and the bubble deformation and centroid movement also weaken.

Fig. 20.

Typical experimental phenomena of bubble collapse in head region with ${\theta }_z=165°$ (Multimedia available online): (a) heart-shaped collapse; (b) elliptical collapse; (c) circular collapse.

Based on the bubble morphological characteristics, three typical cases are distinguished: (i) heart-shaped collapse, (ii) elliptical collapse, and (iii) circular collapse. The primary characteristics of these three cases are summarized below.

In Fig. 20(a), the bubble exhibits a typical heart-shaped collapse with a small dimensionless bubble–hydrofoil distance ($l^{\ast }=1.07$). At the beginning of the collapse, the bubble interface closest to the head of the hydrofoil first shrinks. Then, the bubble interface far away from the hydrofoil also shrinks with a higher velocity, and develops into a collapse jet, with its direction toward the hydrofoil surface. Thus, a typical heart-shaped collapse is observed in Fig. 20(a-8). At the end of the collapse, the jet penetrates the bubble and splits it into two parts of different sizes [i.e., Fig. 20(a-10)]. In addition, the bubble centroid moves toward the surface of the hydrofoil significantly.

In Fig. 20(b), the bubble exhibits a typical elliptical collapse with a medium dimensionless bubble–hydrofoil distance ($l^{\ast }=1.73$). During the collapse process, the bubble interface closest to the hydrofoil head and that far from the hydrofoil shrink at approximately the same velocity. Thus, a typical elliptical collapse is observed in Fig. 20(b-8). At the end of the collapse, a weaker jet is generated and also splits the bubble into two parts of different sizes [i.e., Fig. 20(b-10)]. Meanwhile, the bubble centroid movement weakens.

In Fig. 20(c), the bubble exhibits a typical circular collapse with a large dimensionless bubble–hydrofoil distance ($l^{\ast }=3.11$). During the collapse process, the bubble maintains a circular shape, which means that the contraction velocity of the interface around the bubble is almost uniform. Also, there is almost no noticeable movement of the bubble centroid.

Next, we investigate the anisotropic characteristics via the motion of the bubble contour. Fig. 21 shows the evolution of the bubble contours over time. Five moments are selected to show the evolution of the bubble contour, and the time intervals between them are roughly the same. The most significant movement of the bubble interface occurs far from the hydrofoil, while the weakest movement occurs close to the hydrofoil. As $l^{\ast }$ increases [i.e., from Fig. 21(a) to (c)], the bubble deformation and centroid movement weaken, and the velocity difference at the bubble interface decreases.

Fig. 21.

Evolution of the bubble contours over time. (a) heart-shaped collapse; (b) elliptical collapse; (c) circular collapse.

Then, the theoretically results of the liquid velocity field is also analyzed. Fig. 22 shows the distribution of the liquid velocity in the head region at a dimensionless time ($T^{\ast }$) of 0.75. Two feature points A and B are defined for subsequent analysis, these being the points on the bubble interface that are closest and farthest from the hydrofoil, respectively. As shown in Fig. 22, when the bubble is extremely close to the hydrofoil, the highest liquid velocity is observed at the lower side of the bubble, which is the closest to the head of the hydrofoil. In addition, a low-velocity region is also observed between the bubble and the hydrofoil. As $l^{\ast }$ increases, the low-velocity region expands and the high-velocity region shrinks.

Fig. 22.

Distribution of liquid velocity in head region: (a) l* = 0.77; (b) l* = 1.15; (c) l* = 1.52.

We also analyze how the thickness-related parameter ($\epsilon$) for symmetric hydrofoils and the camber-related parameter ($\delta$) for asymmetric hydrofoils affect the parameter partitioning in the head region. Fig. 23, Fig. 24 show the parameter partitioning for the three cases near symmetric and asymmetric hydrofoils. The photographs in the red dashed boxes show the typical bubble collapse morphology associated with l*. The partition parameter $l^{\ast }$ for cases 1 and 2 increases with increasing $\epsilon$ and decreasing $\delta$, and the partition parameter $l^{\ast }$ for cases 3 and 4 increases with increasing $\epsilon$ and $\delta$.

Fig. 23.

Parameter partitioning for three cases near symmetric hydrofoils: (a)$\epsilon$ = 0.13; (b)$\epsilon$ = 0.27; (c)$\epsilon$ = 0.45. The photographs in the red dashed boxes show the typical bubble collapse morphology associated with l*.

Fig. 24.

Parameter partitioning for three cases near asymmetric hydrofoils: (a) $\delta$ = 120°; (b) $\delta$ = 135°; (c) $\delta$ = 150°. The photographs in the red dashed boxes show the typical bubble collapse morphology associated with l*.

Next, we analyze the velocity of the bubble interface quantitatively to show its deformation characteristics. Fig. 25 shows the bubble interface velocities at points A (v_A) and B (v_B) during the first oscillation period, with points A and B as defined in Fig. 22. As shown in Fig. 25, at the beginning of the bubble growth, the velocity of the bubble interface is the highest and is unaffected by the hydrofoil parameters. The velocity then decreases sharply to zero, reaching the moment of maximum bubble volume. During the collapse process, the velocity increases again, and $v_A$ is always lower than $v_B$. At this point, the impact of the hydrofoil parameters on the interface velocity becomes apparent. Specifically, at the end of the collapse, as $\epsilon$ increases, $v_A$ decreases and $v_B$ increases, and as $\delta$ increases, $v_A$ increases and $v_B$ decreases. In general, the effects of $\delta$ and $\epsilon$ on the bubble interface velocity are opposite to each other.

Fig. 25.

Bubble interface velocities at points A (v_A) and B (v_B) during first oscillation period: (a) and (b) symmetric hydrofoils with$\epsilon$ = 0.13, 0.27, and 0.45; (c) and (d) asymmetric hydrofoils with $\delta$ = 120°, 135°, and 150°.

Then, the influence of the dimensionless distance ($l^{\ast }$) on the roundness of the bubble cross-section at the same moment is analyzed. Fig. 26 shows the roundness ($\varphi$) of bubble cross section at $T^{\ast }=0.94$. The moment at $T^{\ast }=0.94$ represents the typical collapse morphology of the bubble at the final stage of the collapse. As $l^{\ast }$ increases, $\varphi$ also increases, corresponding to the variations in the three typical collapse morphologies shown in Fig. 24. When $l^{\ast }$ is small, the smaller the hydrofoil parameters $\epsilon$ and δ, the smaller $\varphi$ will be. When $l^{\ast }$ is large, $\varphi$ remains at a relatively high level, and the hydrofoil parameters have no significant effect on it.

Fig. 26.

Roundness ($\varphi$) of bubble cross section at $T^{\ast }=0.94$: (a) symmetric hydrofoils with $\epsilon$ = 0.13, 0.27, and 0.45; (b) asymmetric hydrofoils with δ = 120°, 135°, and 150°.

6. Conclusion

Explored herein were the mechanisms by which hydrofoil parameters affect bubble dynamics within a confined space. Experiments involving high-speed photography detailed the typical bubble morphological evolution near different hydrofoils, and the liquid velocity field and Kelvin impulse were analyzed theoretically and compared with the experimental results. The results presented in this paper are applicable to a static, ideal two-dimensional liquid environment, specifically an inviscid, irrotational, and incompressible fluid, and assume the ideal cylindrical motion of the bubble, neglecting the effects of the heat and mass transfer. The main conclusions are given as follows.

• (1)The bubble morphological evolution near a hydrofoil can be categorized into several cases, i.e., arcuate, elliptical, and circular collapses in the tail region, “B”-shaped and heart-shaped collapses in the middle region, and heart-shaped, elliptical, and circular collapses in the head region.
• (2)As the thickness-related ($\epsilon$) hydrofoil parameter increases, the bubble interface velocity and cross-sectional roundness both increase, and the camber-related ($\delta$) hydrofoil parameter have the opposite effect.
• (3)High-velocity regions appear between the bubble and the hydrofoil head and tail endpoints, which explains the bubble interface depressions seen in the experiments. Also, the theoretical Kelvin-impulse direction is highly consistent with the experimental jet angle.

CRediT authorship contribution statement

Junwei Shen: Writing – original draft, Supervision, Formal analysis. Hongbo Wang: Visualization, Validation, Software. Cheng Zhang: Visualization, Methodology, Conceptualization. Yuning Zhang: Writing – review & editing, Supervision, Methodology, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.