Wall vortex induced by the collapse of a near-wall cavitation bubble: Influence of the water surface

Wall vortex occurs when a cavitation bubble oscillates far from a single rigid wall (at a dimensionless standoff distance of ${\gamma }_r>1.3$). This study reveals that introducing a water surface expands the wall vortex regime. A wall vortex in an expanded new regime forms instead of a free vortex at a smaller ${\gamma }_r$ value. Because of the influence of the water surface, a broader jet pierces the bottom of a bubble. This causes the bubbles to expand easily along the wall and form a flat shape during the second cycle. Here an outwards flow forms instead of an upward flow after the bubble recollapses. This study investigates the formation and development of a wall vortex in the new expanded regime via a combination of experiments, numerical simulations, and theoretical modeling. To this end, a theoretical model describing the radial motion $R$ and centroid position $h$ of the bubble between the boundaries is developed using Lagrangian formulation. Two infinite sets of image bubbles are used to satisfy the conditions of the water surface and rigid wall based on image theory. The criteria for the vortex flow patterns are proposed based on the direction of the centroid migration $\dot{h}\left(t_c\right)$ of the bubble at the beginning of the second cycle $t_c$. A free vortex occurs when the upward flow dominates $[\dot{h}\left(t_c\right)>0]$, whereas a downwards flow dominates the wall vortex $[\dot{h}\left(t_c\right)<0]$. A phase diagram of the vortex flows is obtained from the theoretical model and is verified using the experimental results. Numerical analysis reveals that the wall vortex flow with the influence of the water surface contributes to a greater wall shear stress and larger area, thereby increasing the potential for surface cleaning. These findings provide new insights for engineering applications such as ultrasonic cleaning.

Figure 1

Formation of two patterns of the vortex flows induced by the collapse of the cavitation bubbles near a single rigid wall: (a) free vortex at ${\gamma }_r=1$ and (b) wall vortex at ${\gamma }_r=1.5$. The red arrows represent the directions of the flow. The values of $T_c$ are 11.94 ms at ${\gamma }_r=1$ and 12.08 ms at ${\gamma }_r=1.5$, respectively.

Figure 2

Schematic of the experimental setup for observing the behaviors of a spark-induced bubble between a rigid wall and water surface: (a) Experimental setup and (b) definitions of the two nondimensional stand-off distances (${\gamma }_r$ to the rigid wall and ${\gamma }_w$ to the water surface). In (a), the green path represents the light source originating from the pulsed laser.

Figure 3

Imaging system for the motion of a bubble between a water surface and rigid wall. To satisfy the boundary conditions, two infinite sets of image bubbles are used to represent the lowest-order contribution to the potential. The blue bubble represents the original bubble as a source of $q\left(t\right)$, the green and orange image bubbles the two infinite sets of image bubbles, $q\left(t\right)$ the image sources, and $-q\left(t\right)$ the image sinks.

Figure 4

Schematic of the computational modeling process: (a) Computational domain and (b) boundary condition settings. In (b), the size of the calculation domain is set to 240 mm in width and 240 mm in height.

Figure 5

Formation of a free vortex at ${\gamma }_r=1$ and ${\gamma }_w=2$: [(1)–(8)] the first cycle, [(9)–(13)] the second cycle, and [(14)–(20)] the subsequent oscillations of the bubble and migration of the generated free vortex. In this case, experiment $R_{\max }$ = 11.97 mm and $T_c$ = 1.10 ms. The flow induced by the bubble collapse is obtained from the image sequences of the experimental observations and the red arrows represent the directions of the flow. An animation of this case is available in video [48].

Figure 6

Formation of a wall vortex at ${\gamma }_r=1$ and ${\gamma }_w=1$: [(1)–(8)] the first cycle, [(9)–(13)] the second cycle, and [(14)–(20)] the subsequent oscillations of the bubble and migration of the generated wall vortex. In this case, experiment $R_{\max }$ = 11.97 mm and $T_c$ = 1.10 ms. The flow induced by the bubble collapse is obtained from the image sequences of the experimental observations, while the red arrows represent the directions of the flow. An animation of this case is available in video [48].

Figure 7

Experimental results for the equivalent radius $R$ and centroid position $h$ of a spark-induced bubble: (a) free vortex at ${\gamma }_r=1$ and ${\gamma }_w=2$ and (b) wall vortex at ${\gamma }_r=1$ and ${\gamma }_w=1$. The blue squares and orange circles represent the equivalent radius and centroid position, respectively.

Figure 8

Comparisons of the bubble shape evolution between experimental observations and numerical simulations. (a) Free vortex at ${\gamma }_r=1$ and ${\gamma }_w=2$: Experiment $R_{\max }$ = 12.06 mm, $T_c$ = 1.31 ms; simulation $R_{\max }$ = 12.21 mm, $T_c$ = 1.32 ms. (b) Wall vortex at ${\gamma }_r=1$ and ${\gamma }_w=1$: Experiment $R_{\max }$ = 11.97 mm, $T_c$ = 1.10 ms; simulation $R_{\max }$ = 12.20 mm, $T_c$ = 1.12 ms. The experimental observations are shown in the background, and the numerical simulations as blue contours. The contours represent the bubble shape at a volume fraction of 0.5.

Figure 9

Variations in the jet duration ($t_p-t_j$) with ${\gamma }_w$ at ${\gamma }_r=1.5$ obtained from the numerical simulations. The upper right corner shows the definition of the jet duration: from the time of formation ($t_j$) to the time of piercing the bubble bottom ($t_p$). The bubble shapes at the jet formation are shown below at four different ${\gamma }_w$. Here the navy and light blue represent the bubble and liquid, respectively.

Figure 10

Variations in the maximum jet radius $R_{\mathrm{jet}}$ obtained from the numerical simulations with ${\gamma }_w$ at ${\gamma }_r=1.5$. The upper right corner shows the definition of the jet radius. The bubble shapes at the jet piercing are shown below at four different ${\gamma }_w$. Here the navy and light blue represent the bubble and liquid, respectively.

Figure 11

Flow fields of the formation of a free vortex obtained from the numerical simulations at ${\gamma }_r=1$ and ${\gamma }_w=2$. In the figure, (1)–(3) represent the first cycle, (4)–(7) the second cycle, (8) and (9) the vortex flow during subsequent oscillations of the bubble, and (10) a global view of the free vortex. In this case, experiment $R_{\max }$ = 12.21 mm and $T_c$ = 1.32 ms. The vorticity fields and streamlines are shown on the left. The amplitudes of the vorticity are represented by the color bars. The volume fraction fields and velocity vectors are shown on the right. Navy and light blue represent the phases of gas and liquid, respectively.

Figure 12

Flow fields of the formation of a wall vortex in an expanded new regime obtained from the numerical simulations at ${\gamma }_r=1$ and ${\gamma }_w=1$. In the figure, (1)–(3) represent the first cycle, (4)–(7) the second cycle, (8) and (9) the vortex flow during subsequent oscillations of the bubble, and (10) a global view of the wall vortex. In this case, experiment $R_{\max }$ = 12.20 mm and $T_c$ = 1.12 ms. The vorticity fields and streamlines are shown on the left. The amplitudes of the vorticity are represented by the color bars. The volume fraction fields and velocity vectors are shown on the right. Navy and light blue represent the phases of gas and liquid, respectively.

Figure 13

Schematic of the criteria for the vortex flow patterns after the second bubble cycle: (a) $\dot{h}<0$ for wall vortex, (b) $\dot{h}>0$ for the free vortex, and (c) $\dot{h}<0$ for the wall vortex with the influence of the water surface. Frames (1)–(4) represent the beginning, rebound, maximum size, and collapse of the bubble in the second cycle, while frame (5) represents the vortex flow pattern. The blue arrows represent the directions of the flow. Here $R$ is the bubble radius, $h$ the centroid position, and $\dot{h}$ the direction of centroid migration.

Figure 14

Phase diagram of the vortex flow induced by the oscillations of a cavitation bubble between a water surface and rigid wall. The plus and square represent the experimental observations of the free vortex and wall vortex, respectively. The orange and blue areas are obtained by solving the theoretical model. The orange area indicates the free vortex regime. The blue area represents the wall vortex, wherein the navy blue area represents the expanded new regime by the water surface. At ${\gamma }_w<4$, the formation and development of the vortex flow are affected by the constraint of the water surface.

Figure 15

Comparisons of the spatiotemporal distributions of the wall shear stress: (a) free vortex, (b) wall vortex, and [(c) and (d)] wall vortex with the influence of the water surface. The influence of the water surface on the sum of the shear stress is shown in (e) and (f). In the spatiotemporal distributions [(a)–(d)], the first color bar represents the inwards shear stress (towards the axis of symmetry), while the second represents the outwards shear stress (away from the axis of symmetry).

Figure 16

Details of the gray value processes by OpenCV: From the original image to the region filling. The original image is from experiment observations. The red lines represent the symmetric axis of the bubble.

Figure 17

Variations in the bubble radius (a) and wall pressure below the bubble centroid (b) for three different grids at ${\gamma }_r=1.5$ and ${\gamma }_w=1.5$. The three grids are a coarse grid, medium grid, and fine grid with 420 000, 580 000, and 810 000 nodes, respectively.

Figure 18

Comparisons of the theoretical results at ${\gamma }_r=1.5$ and ${\gamma }_w=2.5$ for different numbers of image bubbles: The (a) bubble radius $R$ and (b) centroid position $h$. $k=1$, 2, 3, and 4 represent 5, 9, 13, and 17 image bubbles, respectively. The enlarged areas are shown in green boxes.

Figure 19

Comparison of the bubble radius and centroid position between the numerical simulations, theoretical model, and experimental observations during the first oscillation: (a) ${\gamma }_r=1.5$ and ${\gamma }_w=2.5$ and (b) ${\gamma }_r=2$ and ${\gamma }_w=3$. The open symbols represent the experimental observations, solid lines the theoretical solutions, and dotted lines the simulation results.