Flow fields and vortex dynamics of bubbles collapsing near a solid boundary

When a cavitation bubble oscillates and collapses in the vicinity of a solid boundary (a substrate), it induces intense microconvection in the surrounding liquid and—of high practical importance—directly at the substrate. As the involved flows are fast, highly unsteady, and possess an impressive shear, experiments are difficult and data are scarce. Here, insight into the generation and dynamics of the liquid flows from individual cavitation bubbles collapsing in the vicinity of a solid boundary is provided. Single laser-induced cavitation bubbles (maximum radius around 375 μm) are seeded at precisely defined standoff distances to a substrate by a focused laser pulse. The bubble shape dynamics are imaged by synchronized high-speed cameras from two perpendicular viewing angles. Recording of the shape dynamics is combined with the simultaneous time-resolved measurement of the full flow field on a micrometer-resolution. Measurements employ a high-speed hybrid particle imaging velocimetry and particle tracking velocimetry technique, with a temporal sampling of up to 135 kHz, using fluorescent microparticles as tracers. The time evolution of the unsteady flow field induced by one and the same bubble over a time period much longer than the bubble lifetime is determined. The shear flow at the substrate is analyzed and a liquid transport toward and away from the substrate surface is demonstrated. Depending on the bubble standoff distance, very different flow patterns are observed. The dominant liquid displacement is caused by the long-lived vortex ring being produced during bubble collapse. Most peculiar, the bubble standoff distance determines the sense of direction of the circulation associated with the vortex ring and, consequently, whether the vortex is ejected from the substrate or radially stretches over it. The results are relevant for the understanding of cavitation effects, such as surface cleaning, erosion, and mixing or in biomedical context and may serve as basis for numerical simulations.

Figure 1

Sketch of the high-speed μPIV/PTV setup and the devices for bubble seeding. See the inset in the upper right corner showing a sample image for a geometrical definition of $\mathbf{\gamma }=\frac{\mathit{s}}{{\mathit{R}}_{\max }}$. The laser sheet illumination appears bright and shows up as a vertical line in this side projection (see also the vertical dashed line). It has a thickness of about 140 μm and is adjusted to cut the plane where the laser-induced bubble is created. The solid boundary spans horizontally on top (see horizontal, dashed line).

Figure 2

Simplified setup for the imaging of the shape dynamics.

Figure 3

Time series of two bubbles, one seeded at $\gamma =1.21$, one at $\gamma =1.42$ $(R_{\max }=414\mu \mathrm{m}$ in both cases, $T_{\mathrm{L}}\left(\gamma =1.21\right)=85\mu \mathrm{s},T_{\mathrm{L}}\left(\gamma =1.42\right)=83\mu \mathrm{s}$, camera exposure time: 360 ns). Each bubble is recorded from two perspectives simultaneously by two synchronized high-speed cameras: in bottom view and side view. In bottom view, the view direction is through the transparent solid substrate so that the jet is directed toward the observer. In side view, the solid boundary is located horizontally, below the bubble (dashed line). The boundary location also gets apparent by a mirror image of the gas phase on the substrate. Nondimensionalized times $\tau$ are given with respect to the bubble seeding (plasma generation). Despite the small standoff difference, the two bubbles exhibit very different dynamics after the first collapse. In Video 1 (Supplemental Material [48]) the respective movies from which the images were taken can be found.

Figure 4

Type of vortex ring generated by a collapsing bubble in dependence on the normalized standoff distance $\gamma$: Each data point on the upper line indicates the observation of a “free vortex” that is encountered in the range $0.5\le \gamma \le 1.30$. The “wall vortex” (data points on the lower line) is observed for $\gamma \ge 1.36$. A very sharp transition between the two flow patterns is present. Only at some intermediate distance $\left(1.30<\gamma <1.36\right)$, and for very small standoffs $(\gamma \lesssim 0.5)$, no vortex ring seems to be created as no directed motion of the bubble fragments is observed. Instead bubble fragments rather “float” around the collapse location (data points on middle line). Maximum bubble radii ranged from $R_{\max }=300\mu \mathrm{m}$ to about $450\mu \mathrm{m}$. For the two observations indicated by the arrows, high-speed imaging series of the gas phase dynamics are given in Fig. 3.

Figure 5

Time evolution of the velocity field of a free vortex generated by a collapsing bubble $(\gamma =1.21;T_{\mathrm{L}}=68$ μs; $R_{\max }=325\mu \mathrm{m}).$ The solid boundary extends along $y=0$. Times indicated in the top left corner of each frame are given with respect to bubble seeding (at $t=0\mu \mathrm{s})$ in absolute values and in nondimensionalized form $\tau =t/T_{\mathrm{L}}$.The motion and dynamics of the gas phase during the velocity measurement are overlaid up to the frame at $\tau =2.96$, thereafter the gas is mainly dissolved. The local velocity is given by the arrow length, the respective scale for each frame is shown in the top right corner. Velocities are integrated over a time span of $\mathrm{\Delta }{T}_{\mathrm{PIV}}=25\mu \mathrm{s}$ around the times given $(\pm 12.5$ μs).

Figure 6

Trajectory of the free ring vortex, characterized by the travelled vertical distance $y_v$ over time. For later use of the translation velocity, a polynomial fit (solid line) is shown.

Figure 7

Time evolution of the velocity field of a bubble generating a free vortex, situation similar to Fig. 5 but captured at a higher frame rate of 135 000 fps $(\gamma =1.1$; $R_{\max }=340\mu \mathrm{m}$; $T_{\mathrm{L}}=75\mu \mathrm{s}$; $T_{\mathrm{L}2}=131\mu \mathrm{s}-T_{\mathrm{L}}=56\mu \mathrm{s})$. The time elapsed since bubble generation is indicated in absolute values and normalized to $T_{\mathrm{L}}$. Frames proceed column-wise. $\mathrm{\Delta }{T}_{\mathrm{PIV}}=7.4\mu \mathrm{s},T_{\mathrm{PIV}}^{\mathrm{e}}=5.5\mu \mathrm{s}$.

Figure 8

Time evolution of the velocity field of a wall vortex (the solid boundary extends along $y=0)$ generated by a collapsing bubble $(\gamma =1.67;T_{\mathrm{L}}=61\mu \mathrm{s}$; $R_{\max }=310\mu \mathrm{m}).$ The motion and dynamics of the gas phase during the velocity measurement are overlaid up to $\tau =2.75$, thereafter the gas is mainly dissolved. Times indicated in the frames are given with respect to bubble seeding (at $t=0\mu \mathrm{s})$. Time sequence row-wise. Velocities are integrated over a time span of 25 μs around the times given.

Figure 9

Time evolution of the velocity field of a bubble generating a wall vortex resolved at 135 000 Hz $\left(\gamma =1.75,T_{\mathrm{L}}=80\mu \mathrm{s},R_{\max }=420\mu \mathrm{m}\right)$. At $t=115\mu \mathrm{s}$ already a circular flow pattern is visible. Then between $t=136\mu \mathrm{s}$ and $144\mu \mathrm{s}$ the wall vortex has clearly emerged at about $y=200\mu \mathrm{m}$. Times $t$ and $\tau$ of frames are given in top left corner in microseconds and normalized form below. Frames proceed from top to bottom and then left to right.

Figure 10

Left: Color coded vorticity $\omega$ of the velocity field from the ring vortex of Fig. 5 (shown here at $t=451$ μs, $\gamma =1.21)$. The dashed white line depicts the path C of the integral that can be used to calculate the circulation. For more details on the determination of the circulation and vorticity see the Appendix pp3. The dashed black line depicts the circular sector within which the data points are considered for the Gaussian fit for the right image. The black lines indicate the radial coordinate $a$ and the ring vortex diameter $D_v$. Right: Measurement data of the vorticity (from the circular sector of the left graphic) with Gaussian fit around the vortex core.

Figure 11

Left: Gaussian vorticity profiles shown at four selected times (fits). Right: Time evolution of the circulation $\mathrm{\Gamma }\left(t\right)$ of the free vortex (determined from the Gaussian vorticity profiles). The times A, B, C, E (see left figure), and D (see Fig. 10, right) for which the vorticity profiles are plotted are indicated.

Figure 12

Comparison of the self-induced ring vortex velocity $U_{\mathrm{\Gamma }}\left(t\right)$ assuming Gaussian vorticity profiles (for $\gamma =1.21)$ compared to the measured velocity (the translation velocity of the vorticity maximum, derived from the fit of Fig. 6). Before $t\cong 210\mu \mathrm{s}$ the vortex is still in the stage of formation (masked space), thereafter the ring vortex has completely developed, and predicted translation velocity and measured velocity correspond well.

Figure 13

Circulation $\mathrm{\Gamma }\left(t\right)$ of the wall vortex of Fig. 8 (determined by the path integral).

Figure 14

Lagrangian ink map of a free vortex derived from the time-resolved flow field that was obtained experimentally by μPIV/PTV. The bubble is generated at $\gamma =1.21$ $(t=0)$, the boundary extends horizontally along $y=0$. Absolute times $t$ and nondimensionalized times $\tau$ are given in each frame. In the first image $\left(t=0\right)$ the initial ink configuration is shown: horizontal, colored layers from white (top) to dark blue (bottom). Also, the bubble shape at its maximum expansion is depicted (the maximum expansion is reached later, at $t=34\mu \mathrm{s})$. Along with the formation and migration of the free vortex ring, liquid is ejected from the boundary layer into the bulk. Also a weaker liquid displacement conveyed by a flow annularly spreading along the substrate is visible. See also Video 2 of the Supplemental Material [48].

Figure 15

Last frame of the ink displacement series of Fig. 14. Indicated is the bubble shape at maximum expansion and a sketch of the dominant directions of liquid displacement (white arrows: away and along the substrate, black toward the substrate).

Figure 16

Vertical ink map for the bubble of Fig. 8 $\left(\gamma =1.67\right)$ generating a wall vortex. The boundary is located horizontally at $y=0$. Absolute times $t$ and nondimensionalized times $\tau$ are given in each frame. Two main streams are responsible for the net displacement and mixing: (1) Along with the jet, fluid is pushed from the bulk toward the boundary (white-yellowish, vertical tip). (2) Along with the radially spreading vortex ring, fluid is pushed horizontally outward (from the symmetry axis) over the substrate. Thereby, the boundary layer lifts off and winds up with the vortical flow (dark blue “spiral”). Times respective to bubble seeding are given in each frame. Initial situation at $t=0$: horizontal, colored layers from white-yellow (top) to dark blue (bottom) as in Fig. 14.

Figure 17

Simplified sketch of the flows generated during the second bubble collapse that are involved in the production of either a free or a wall vortex. (a) Free vortex: The flat, elongated bubble collapses mainly from the outer torus, high pressures occur at the center and the axial jet quickly stagnates. (b) Wall vortex: The axial jet pierces through the bubble and impacts on the substrate before the gas phase.